text:
\section{Introduction}

\subsection{Statement of results}
Let $\Omega\subset\re^n$, $n\ge 2$, be a smooth bounded domain, and $u(x)\in C^\infty(\ov\Omega)$ be the smooth solution of the quasilinear boundary value problem
\eqal{\label{eq}
\nabla\cdot(a(u,\nabla u)\nabla u)&=0,\qquad x\in\Omega,\\
u|_{\pd\Omega}&=f,
}
where $a(s,p)\in C^\infty(\re\times\re^n)$ satisfies standard conditions to make this BVP wellposed, see \cite{MU} and \cite{GT}.  We assume summation over repeated indices.

\medskip
\noindent
\textit{Ellipticity:}
\eqal{\label{aij}
0<\lambda(s,p)|\xi|^2\le &\,a_{ij}(s,p)\xi_i\xi_j,\quad (s,p,\xi)\in\re\times\re^n\times(\re^n\setminus\{0\}),\\
&a_{ij}:=a\,\delta_{ij}+\frac{1}{2}(a_{p_i}p_j+a_{p_j}p_i).
}
\textit{Coercivity:}
\eqal{
\label{coer}
\lambda(s,p)\ge \lambda_0(|s|)>0,\\
a(s,p) \ge 1
}
\textit{Growth:}
\eqal{
\label{grow}
|p||\nabla_p a|+|a|\le \mu_0(|s|),\\
(1+|p|)|a_s|\le\mu_0(|s|)|p|.
}
The bounds $\lambda,\mu_0,\lambda_0
$ and domain $\Omega$ are known quantities.  Positive functions $\lambda_0$ and $\mu_0$ are non-increasing and non-decreasing in $|s|$, respectively.
We also assume known bounds on the $C^k(K)$ seminorms of $a$ for all $k\ge 0$ and all compact subsets $K$ of the domain of $a$.

\medskip
Now, the Dirichlet-to-Neumann (DN) map is defined by
\eqal{\label{DN}
f\mapsto \Gamma_a(f)
&=a(u,\nabla u)\frac{\pd u}{\pd \nu}dS,
}
where $dS$ is the Euclidean area form on $\pd\Omega$, and $\frac{\pd u}{\pd \nu}=\nu\cdot\nabla u$ is the Euclidean normal derivative.

\medskip
Under no additional assumptions, we can recover $a$ from $\Gamma_a$ along sufficiently small gradient values.

\begin{thm}\label{thm:open}
If $\Gamma_a=\Gamma_{\ti a}$, then $a(s,p)=\ti a(s,p)$ on an open set:
$$
\{(s,p):|p|< \Pi(s)\},
$$
where $\Pi:\re\to(0,\infty)$ is a positive continuous function which depends on the known quantities.
\end{thm}

\medskip
We next recover two classes of $a(s,p)$ along large gradients.  The first follows immediately from Theorem \ref{thm:open}.

\begin{thm}\label{thm:analytic}
Let $a(s,p)$ and $\ti a(s,p)$ be real-analytic.  If $\Gamma_a=\Gamma_{\ti a}$, then $a(s,p)\equiv \ti a(s,p)$.
\end{thm}

Theorem \ref{thm:analytic} extends the result of \cite{MU}, which assumed conditions on the holomorphic extension of real analytic $a(s,p)$ in order to solve a complex-valued BVP.

\begin{thm}\label{thm:decay}
Let $a$ and $\ti a$ satisfy two additional conditions:

\medskip
1. Uniform elliptic lower bound:
\eqal{
\label{uni}
a_{ij}(s,p),\quad \ti a_{ij}(s,p)\ge \lambda_0>0,\qquad (s,p)\in\re\times\re^n,
}
where the inequality is in the quadratic form sense, and $\lambda_0$ is constant.

\medskip
2. Decay along large gradients:
\eqal{
\label{decay}
|a_s(s,p)|,\quad |\ti a_s(s,p)|\le  \frac{\ep_0}{|p|},\qquad (s,p)\in\re\times\re^n,
}
for some sufficiently small $\ep_0>0$ depending on $\Omega$ and $\lambda_0$.

\medskip
If $\Gamma_a=\Gamma_{\ti a}$, then $a(s,p)\equiv \ti a(s,p)$.
\end{thm}

This seems to be the first example of global uniqueness of a non-analytic conductivity depending on $(u,\nabla u)$.

\medskip
Finally, as shown in Remarks \ref{rem:n3}, \ref{rem:bsym}, and \ref{rem:n2}, the above results hold for non-homogeneous conductivities $a=a(x,s,p)$ as well, provided the ``$x$-dependent part" is either known  (i.e. is the same for both $a$ and $\ti a$) or symmetric along a direction, say $(0,\dots,0,1)\in\re^n$.
We assume the known quantities in \eqref{aij}, \eqref{coer}, and \eqref{grow}
 are unchanged and do not depend on $x$, for simplicity.

\begin{cor}\label{cor:x}
If $n\ge 3$, suppose also $\Omega$ is convex and $a$ and $\ti a$ are of the form
\eqal{
a(x,s,p)=F(x,s,p,b(x^1,\dots,x^{n-1},s,p)),\\
\ti a(x,s,p)=F(x,s,p,\ti b(x^1,\dots,x^{n-1},s,p)),
}
where $t\mapsto F(x,s,p,t)$ is injective for each fixed $(x,s,p)$.  (If $n=2$, remove convexity of $\Omega$ and assume $b$ and $\ti b$ are functions of $(s,p)$ only).

\medskip
Then the statements of Theorems \ref{thm:open}, \ref{thm:analytic}, and \ref{thm:decay} are true with replacements of $(s,p)$ by $(x,s,p)$, with $x\in\Omega$.
\end{cor}

The fully general case where $b=b(x,s,p)$ is unclear to the author and may require new methods.

\begin{comment}
\begin{thm}\label{thm:all}
Let $a$ and $\ti a$ satisfy one of the following cases:

\medskip
Case 1: $a$ and $\ti a$ are real analytic.

\medskip

Case 2: $a$ and $\ti a$ satisfy the decay condition
\eqal{
\label{decay}
|a_s|,|\tilde a_s|\le \ep(\Omega)\,\frac{\lambda(s,p)}{|p|},\quad (s,p)\in\re\times\re^n.
}

\medskip
Case 3: $a$ and $\tilde a$ are symmetric in one direction:
\eqal{
\label{tsym}
a_{p_n}(s,p) = \ti a_{p_n}(s,p)=0.
}

\medskip
\noindent
If $\Gamma_a=\Gamma_{\ti a}$, then $a\equiv \ti a$.
\end{thm}

\end{comment}

\subsection{Discussion}
The classical Calder\'on problem considers an isotropic linear elliptic equation and its Dirichlet-to-Neumann map
\eqal{
\label{cald}
\nabla\cdot (a(x)\nabla u)=0,\\
\Gamma_a:u|_{\pd\Omega}\mapsto a(x)\frac{\pd u}{\pd \nu}\Big|_{\pd\Omega}.
}
The problem is to determine $a(x)$ from $\Gamma_a$.  See \cite{U} for a survey of this problem and its extensions.  The matrix extension $a(x)=a_{ij}(x)$, or the anisotropic Calder\'on problem, is still unsolved in general for dimension three or higher.

\medskip
Recent work has featured the quasilinear generalization $a=a(x,u,\nabla u)$.  The case $a=a(x,u)$ was considered in \cite{S1}, using a linearization technique similar to that originally introduced in \cite{I1}.  This was extended in \cite{SU} to the anisotropic case $a=a_{ij}(x,u)$ (matrix).  See \cite{HS} for analysis in the case $a=a_{ij}(x,\nabla u)$ in dimension two, and \cite{KN} for the recovery of a Taylor polynomial of $a=a_{ij}(x,\nabla u)$.  The linearization technique was also applied to the semilinear equations $\Delta u+a(x,u)=0$ and $\Delta u+a(x,\nabla u)=0$ by \cite{IS} and \cite{S2}, respectively.  More recently, a multilinearization technique was developed for $\Delta_gu+a(x,u)=0$ in \cite{FO,KU,LLLS}, where $a$ is analytic in $u$.  The equation $\Delta u+f(x)|\nabla u|^2+V(x,u)=0$ with $V$ analytic was determined in the partial data case in \cite{KU1} using multilinearization.
In \cite{I2}, the equation $\Delta u+c(u,\nabla u)=0$ was considered using singular solutions of the linearized equation which concentrate at the boundary.  The gradient structure was used for the quasilinear equation $\nabla\cdot(a(u)\nabla u)+c(x)u=0$ with lower order term in \cite{EPS}.

\medskip
In the work \cite{MU} by Mu\~noz and Uhlmann, attention was brought to the quasilinear case $a=a(u,\nabla u)$, and results were obtained assuming conditions on the holomorphic extension of $a$.  These conditions ensure that linearization is possible around complex-valued affine solutions.  The question is raised whether we can allow for \textit{arbitrary} $a(s,p)$ real analytic, $a(s,p)$ smooth but \textit{non-analytic}, and \textit{non-homogeneous} $\pd a(x,s,p)/\pd x\neq 0$.

\medskip
The purpose of this work is to extend the results of \cite{MU} to several new classes of quasilinear conductivities:

\medskip
1. $a(u,\nabla u)$ real analytic, without additional hypotheses, Theorems \ref{thm:open} and \ref{thm:analytic}.

\medskip
2. $a(u,\nabla u)$ smooth, assuming additional conditions \eqref{uni} and \eqref{decay}, Theorem \ref{thm:decay}

\medskip
3. Non-homogeneous cases $a=a(x,u,\nabla u)$ with known $x$ part, Corollary \ref{cor:x}.

\medskip
An apparent \textit{a priori} difficulty of the inverse problem for $a=a(u,\nabla u)$ (and truly one for general $a=a(x,u,\nabla u)$) is that the linearized inverse problem is anisotropic.  Not only is the anisotropic problem not understood in dimension three or higher, but the matrix $a_{ij}(x)$ can only be recovered up to pushforward by a boundary-preserving diffeomorphism; see \cite{SU} and \cite{S2} for similar difficulties.  In the $p$-Laplace case $a=A(x)|\nabla u|^{p-2}\nabla u$, the ``nonlinear part" of the quasilinearity is known, unlike for equation \eqref{eq}, and boundary determination was possible in \cite{SX} using explicit Wolff-type oscillatory solutions, with full recovery using a monotonicity identity \cite{GKS}.  If instead the anisotropic and nonlinear part is small, then one may linearize near the zero solution as in \cite{KN} and recover the $x$-dependent Taylor coefficients using linear theory.  This is similar to the idea of the multilinearization approach \cite{FO,KU,KU1,LLLS} for which dependence on $u$ or $\nabla u$ is analytic.  Constant solutions were used for linearizing the conductivity equation in \cite{SU}, and complex-valued affine solutions were used for linearization in \cite{MU}.
However, it is not clear what non-constant, explicit solutions quasilinear equation \eqref{eq} has in full generality.  If $a(s,p)$ is smooth but not real analytic, then complex-valued affine solutions no longer make sense.

\begin{comment}
\medskip
One difficulty with this problem is that there is not currently a theory for the linearized inverse problem, which is anisotropic.  Formally, if $u=u[f]$ is the solution of \eqref{eq}, the derivative
$$
v=\frac{d}{d\ep}\Big|_{\ep=0}u[f+\ep h]
$$
solves the anisotropic boundary value problem
\begin{align*}
\nabla\cdot (A(x)\cdot \nabla v)+b(x)\cdot\nabla v+c(x)v=0,\\
v|_{\pd\Omega}=h,
\end{align*}
where $A_{ij}=a_{ij}$ is in \eqref{aij}.  One would hope that if the linearized DN maps $\Gamma_a[u]=\Gamma_{\ti a}[\ti u]$ are equal, then $A(x)=\ti A(x)$ as well.  In fact, we could only hope for equality up to pushforward by a boundary-preserving diffeomorphism, see \cite{SU}. But unless the matrix $A(x)$ is real analytic or of a special product-type form, see e.g. \cite{DKSU}, such conclusions are not known for dimension $n\ge 3$.  Equations of the isotropic form
\eqal{
\Delta u+a(x,u)+b(x,\nabla u)=0,
}
for which the linearized inverse problems are solvable, have been successfully considered using the linearization method, see e.g. \cite{IS} and \cite{S2}.  See also \cite{GKS} for the $p$-Laplace case $a=a(x)|\nabla u|^{p-2}$, using a different method when more structure is available.

\medskip
Moreover, even in the solvable dimension $n=2$ case, it is not clear which solution $u=u[f]$ to linearize the equation about.  The equation
$$
\nabla\cdot (A(x,u)\cdot\nabla u)=0
$$
considered in \cite{S1} and \cite{SU} has constant solutions.  The equation
$$
\nabla\cdot (A(\nabla u)\cdot \nabla u)=0
$$
has affine solutions $u(x)=\al\cdot x+\beta$, and it was shown in \cite{MU} that similar solutions exist for $a=a(u,\nabla u)$ if $a$ has a holomorphic extension.  However, in general, the existence of explicit solutions beyond constants is unclear, so the diffeomorphism obstruction prevents a complete solution to the inverse problem; see \cite{HS} for a solution when one of $A(x,\nabla u)$ or $\ti A(x,\nabla u)$ depends only on $x$ or $\nabla u$.

The strategy to recover nonlinearities in elliptic inverse problems is to linearize the equation about a solution as in [I1], then invoke the theory of inverse problems of linear elliptic equations, and finally vary the original solution.  See [U] for progress in the linear direction.  When the principal part of the nonlinear equation is Laplacian $\Delta$ (isotropic), this was successful in [IS] for semilinear equations $\Delta u+a(x,u)=0$ and in [S] for quasilinear equations $\Delta u+a(x,\nabla u)=0$, invoking the results using the complex geometrical optics construction.  In [I2], the procedure needed modification for the quasilinear equation $\Delta u+c(u,\nabla u)=0$, and boundary determinination theory was used instead (singular solutions concentrating at the boundary).  We also mention several works [] on quasilinear equations which recover $x$-dependent quantities.

\medskip
Much less is known when the principal part is itself quasilinear and unknown, since the linearized problem is anisotropic.  For the quasilinear equation $\nabla \cdot (A(x,u)\cdot\nabla u)=0$, the matrix $A(x,s)$ was recovered [SU] for dimension two or $A$ analytic.  For the equation $\nabla\cdot (A(x,\nabla u)\cdot\nabla u)=0$, it was shown [HS] that a matrix of the forms $A=A(p),A(x)$ could be distinguished using the DN map.  For the isotropic equation $\nabla\cdot(a(u,\nabla u)\nabla u)=0$, scalar $a(s,p)$ was recovered in [MU] for $a$ verifying analytic hypotheses.
\end{comment}

\medskip
Our approach is to use classical boundary determination theory, which is well understood even in the anisotropic case.  We first linearize the equation around any solution with a prescribed boundary jet, then recover the tangential part of the linearized conductivity matrix \eqref{aij} using boundary determination theory.  Next, the isotropic structure of the linearized conductivity matrix yields a solvable algebraic system for the conductivity evaluated at the jet.
According to the comparison principle, we can use logarithmic barriers to prescribe boundary jets on a small open subset, which will complete the proof of Theorem \ref{thm:open}.   For Theorem \ref{thm:decay}, decay condition \eqref{decay} ensures that exponential barriers exist, hence solutions with any prescribed boundary jet in $\re\times\re^n$.

\medskip
This work is organized as follows.  In Section \ref{sec:lin}, we show that the linearized BVP is wellposed and has a well-defined DN map.  In Section \ref{sec:jet}, we use the comparison principle to construct solutions with prescribed boundary jets.  In Section \ref{sec:geo}, we geometrically reformulate the linearized problem, which allows us to invoke known boundary determination theory in Section \ref{sec:bdy}.  In Section \ref{sec:proof}, we complete the proofs of Theorems \ref{thm:open} and \ref{thm:decay}.

\section{Preliminaries}

\subsection{The linearized problem}
\label{sec:lin}

For arbitrary solutions to the quasilinear equation \eqref{eq}, the linearized Dirichlet problem centered at that solution may not be solvable.  Under a small gradient assumption or the decay assumption \eqref{decay}, we will show the uniqueness of weak solutions to the linearized equation, and standard techniques will again apply after invoking higher regularity.  Note that the non-homogeneous case $a=a(x,u,\nabla u)$ applies here without change as well.

\medskip
We denote by $u=u[f]$ the solution of the BVP \eqref{eq}.  If $f,h\in C^\infty(\pd\Omega)$, then formally, we let
\eqal{
\label{vdef}
v(x):=\frac{d}{dt}\Big|_{t=0}u[f+t h](x)=\lim_{t\to 0}\frac{u[f+th](x)-u[f](x)}{t}
}
solve the linearized Dirichlet problem
\eqal{\label{eqL}
\pd_i(a_svu_i+a_{p_j}v_ju_i+av_i)&=0,\\
v|_{\pd\Omega}=h.
}
The linearized DN map:
\eqal{\label{DNL}
\Gamma_a[u](h):=\frac{d}{dt}\Big|_{t=0}\Gamma_a(f+t h)=(a_sv\frac{\pd u}{\pd \nu}+a_{p_j}v_j\frac{\pd u}{\pd \nu}+a\frac{\pd v}{\pd \nu})dS.
}

\medskip
For brevity, we say that \textit{the linearization of \eqref{eq} exists at $u$} if the limit \eqref{vdef} is well defined in $C^{1}(\Omega)$ and the Dirichlet problem \eqref{eqL} for $v$ is uniquely solvable in $W^{1,2}(\Omega)$.  If $\Gamma_a=\Gamma_{\ti a}$ and the linearization exists at $u$ and $\ti u$, it then follows that $\Gamma_a[u]=\Gamma_{\ti a}[\ti u]$

\medskip
To show that the linearization of \eqref{eq} exists, we first specialize to solutions $u$ with small gradients, under no additional assumptions on $a(s,p)$.

\begin{prop}
\label{prop:lin_small}
Given $s\in\re$, there exists $\Pi_1>0$ sufficiently small such that if $f\in C^\infty(\pd\Omega)$ satisfies
$$
\|f-s\|_{C^{2,\al}(\pd\Omega)}\le \Pi_1,
$$
then the linearization of \eqref{eq} exists at $u=u[f]$.  Here, $\al\in(0,1)$ is as in \cite[Theorem 2.10]{MU} and depends on the known quantities and $s$.
\end{prop}

To linearize \textit{near} constant solutions, we need to verify the solution operator is Lipschitz continuous \textit{at} constants.
\begin{lem}
Given $k\in\re$ and $h\in C^{2,\al}(\pd\Omega)$, there exists $\al\in(0,1)$ sufficiently small such that the following estimate holds:
\eqal{
\label{C2ak}
\|u[k+h]-k\|_{C^{2,\al}}\le C(k,\|h\|_{C^2})\|h\|_{C^{2,\al}}.
}
\end{lem}

\begin{proof}
Letting $f=k+h$ and $u=k+w$ in \eqref{eq}, we obtain the BVP
\eqal{
\label{eqt}
\nabla\cdot\left(a(k+w,\nabla w)\nabla w\right)=0,\\
w|_{\pd\Omega}=h.
}
The structural conditions \eqref{aij}, \eqref{coer}, and \eqref{grow} are still satisfied, with lower and upper bounds now also depending on a parameter $k$.  It was shown in \cite{MU} that \eqref{eq} has an \textit{a priori} $C^{1,\al}$ estimate in terms of $\|f\|_{C^2}$ for some $\al\in(0,1)$ depending on $a$ and $\|f\|_{C^2}$, which means each coefficient in the nondivergence form of \eqref{eqt},
\eqal{
a_{ij}(k+w,\nabla w)w_{ij}+(a_w(k+w,\nabla w)w_i)w_i=0,
}
has a $C^\al$ estimate also in terms of $\|f\|_{C^2}$.  Therefore, the linear Schauder $C^{2,\al}$ estimate \cite[Theorem 6.6]{GT} yields
\eqal{
\label{C2a}
\|u\|_{C^{2,\al}(\Omega)}\le C(a,\Omega,\|f\|_{C^2(\pd\Omega)})\|f\|_{C^{2,\al}(\pd\Omega)}.
}
In the case of \eqref{eqt}, the same estimate is therefore true, with $C(a,\Omega,\|f\|)$ replaced by $C(k,a,\Omega,\|h\|)=:C(k,\|h\|)$, and $\al$ depending also on $k$ and $\|h\|_{C^2}$.  Substitution of $w=u-k$ into the above completes the proof.
\end{proof}

We also need the continuity of the solution operator near \textit{any} small solution, which we record for completeness.  Given $M>0$ and the sufficiently small $\al=\al(M)$ of \eqref{C2a}, we give the subset $B_M=\{f\in C^{2,\al}(\pd\Omega):\|f\|_{C^2(\pd\Omega)}< M\}$ the $C^{2,\al}(\pd\Omega)$ norm.

\begin{lem}
\label{lem:cont}
For each $M>0$ and $\beta\in(0,\al(M))$, the operator $B_M\to C^{2,\beta}(\Omega)$ given by $f\mapsto u[f]$ is continuous.
\end{lem}

\begin{proof}
If not, then there exists a sequence $f_k\in C^{2,\al}(\pd\Omega)$ with $\|f_k\|_{C^2}\le M$ converging to $f\in C^{2,\al}(\pd\Omega)$ but with $\|u[f_k]-u[f]\|_{C^{2,\beta}(\Omega)}\ge 1$, say.  By \eqref{C2a}, the solutions $u[f_k]$ are uniformly bounded in $C^{2,\al}(\ov\Omega)$, so the Arzela-Ascoli theorem yields a subsequence, also named $u[f_k]$, which converges in $C^{2,\beta}(\ov\Omega)$ to some $v$.  It follows that $\|v-u[f]\|_{C^{2,\beta}(\Omega)}\ge 1$.  However, since each $u[f_k]$ solves \eqref{eq} with boundary data $f_k$, sending $k\to\infty$ in \eqref{eq} shows that $v$ solves \eqref{eq} as well with boundary data $f$.  By the comparison principle \cite[Theorem 10.7]{GT} for quasilinear divergence form operators, it follows that $C^{2,\beta}$ solutions are unique, hence $v=u[f]$, a contradiction.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:lin_small}]
Defining $t$-dependent quantities,
\eqal{
u(x;t):=u[f+th](x),\quad v(x;t):=\frac{u(x;t)-u(x;0)}{t},\\
a(x;t):=a[u(x;t)],\quad u(;\sigma;t)=u(;0)+\sigma(u(;t)-u(;0)),
}
with $a[u]:=a(u,\nabla u)$, we take a ``discrete derivative":
\eqal{
a(;t)u_i(;t)-a(;0)u_i(;0)&=ta(;t)v_i(;t)+(a(;t)-a(;0))u_i(;0)\\
&=tav_i+ta_s(;t)v(;t)u_i+ta_{p_j}(;t)v_j(;t)u_i,
}
where, by the fundamental theorem of calculus,
\eqal{
a_s(;t):=\int_0^1a_s[u(;\sigma;t)]d\sigma,\quad a_{p_j}(;t):=\int_0^1a_{p_j}[u(;\sigma;t)]d\sigma.
}
We see that $v(;t)$ then solves the Dirichlet problem
\eqal{
\label{probt}
\pd_i\left(a_s(;t)v(;t)u_i(;0)+a_{p_j}(;t)v_j(;t)u_i(;0)+a(;t)v_i(;t)\right)=0,\\
v(;t)|_{\pd\Omega}=h,
}
with associated Cauchy data (DN map)
\eqal{
\left(a_s(;t)v(;t)\frac{\pd u}{\pd \nu}(;0)+a_{p_j}(;t)v_j(;t)\frac{\pd u}{\pd\nu}(;0)+a(;t)\frac{\pd v}{\pd\nu}(;t)\right)dS.
}

\medskip
\textbf{Step 1:}

We verify the Dirichlet problem \eqref{probt} for $v(;t)$ has a unique weak solution in $W^{1,2}(\Omega)$ which is uniformly bounded in $t$.  Let $w\in W^{1,2}_0$ solve
\eqal{
\label{eqLw}
\pd_i\left(a_s(;t)wu_i(;0)+a_{p_j}(;t)w_ju_i(;0)+a(;t)w_i\right)=0.
}
Multiplying \eqref{eqLw} by $w$ and integrating by parts, we obtain
\eqal{
\int_\Omega a_{ij}(;t)w_iw_j\,dx=-\int_\Omega a_s(;t)u_i(;0)ww_i\,dx,
}
where we defined
\eqal{
a_{ij}(;t):=a(;t)\delta_{ij}+\frac{1}{2}\int_0^1 u_i(;0)a_{p_j}[u(;\sigma;t)]+u_j(;0)a_{p_i}[u(;\sigma;t)]\,d\sigma.
}
For $|t|\le 1$, estimate \eqref{C2a} shows $u(;t)$ is uniformly bounded in $C^{2,\al}$ with respect to $t$, for $\al$ sufficiently small.  Moreover, Lemma \ref{lem:cont} shows that $t\mapsto u(;t)$ is continuous in $C^{2,\beta}$ if $\beta\in(0,\al)$, so by the smoothness of $a$, we can write $a_{ij}(;t)=a_{ij}(;0)+r_1(t)$, where remainder $r_1(t)$ vanishes in $C^{1,\beta}$ as $t\to 0$.  By \eqref{grow}, we can therefore suppose $t$ is sufficiently small so that
\eqal{\label{aijbound}
a_{ij}(;t)\xi_i\xi_j\ge \frac{1}{2}\lambda_0(|u(;0)|)|\xi|^2,\qquad \xi\in\re^n.
}
Similarly, we write $a_s(;t)=a_s(;0)+r_2(t)$, and recalling \eqref{grow}, we suppose $t$ is sufficiently small so that
\eqal{\label{asbound}
|a_s(;t)|\le 2\mu_0(|u(;0)|).
}
The Cauchy-Schwarz inequality then yields
\eqal{
\label{ineq4}
\int_\Omega\lambda_0(|u(;0)|)|Dw|^2\,dx\le 4\int_\Omega \frac{\mu_0^2}{\lambda_0}(|u(;0)|)|Du(;0)|^2w^2\,dx
}
Since $Du=D(u-s)$, the estimate \eqref{C2ak} implies the gradient is small:
\eqal{
\|Du(;0)\|_{L^\infty}\le C(s)\|f-s\|_{C^{2,\al}}\le C(s)\Pi_1.
}
If $\Pi_1\le 1$, then $\|f\|_{C^{2,\al}}\le |s|+C(s)$, and $C^{2,\al}$ estimate \eqref{C2a} yields
\eqal{
\|u(;0)\|_{L^\infty}\le C(s),\quad \lambda_0(|u(;0)|)\ge \lambda_0(C(s)),\quad \mu_0(|u(;0)|)\le \mu_0(C(s)),
}
for another $C(s)$.  Substituting these and the Poincar\'e inequality for $w\in W^{1,2}_0(\Omega)$ into \eqref{ineq4} yields
\eqal{
\int_\Omega|Dw|^2\,dx\le \frac{\mu_0^2}{\lambda_0^2}(C(s))\cdot C(s,\Omega)\cdot\Pi_1^2\int_\Omega|Dw|^2\,dx.
}
Choosing $\Pi_1$ small depending on $s,\Omega$ confirms that $w=0$.  The Fredholm alternative \cite[Theorem 5.3]{GT} shows that Dirichlet problem \eqref{probt} has a unique solution $v(;t)\in W^{1,2}(\Omega)$, and \cite[Corollary 8.7]{GT} yields the $W^{1,2}$ estimate
\eqal{
\|v(;t)\|_{W^{1,2}(\Omega)}\le C(\eqref{probt},\Omega)\|h\|_{W^{1,2}(\Omega)},
}
where $C(\eqref{probt},\Omega)$ indicates (uniform) dependence on the coefficients of the differential operator in \eqref{probt}.  For small $t$, the coefficients and their derivatives have controlled $t$ dependence in the $C^\beta$ norm, so we conclude that $v(;t)$ has a uniform-in-$t$ $W^{1,2}$ bound.  Moreover, since $f,h\in C^\infty$, standard Schauder bootstrapping yields higher regularity $C^{k,\al}(\pd\Omega)\to C^{k,\al}(\Omega)$ estimates for $u(;t)$ as in \eqref{C2a}, so the $t$ dependence of the coefficients in \eqref{probt} is uniform in any $C^k$ norm.   Therefore, the higher regularity $W^{k+2,2}(\Omega)$ estimate \cite[Theorem 8.13]{GT} combined with Sobolev embeddings yield uniform-in-$t$ estimates for $v(;t)$ in any $C^\ell$ norm.

\medskip
\textbf{Step 2:}

We now show $v(;t)$ converges to $v$ in $C^1$, where $v$ is the unique solution of Dirichlet problem \eqref{eqL}.  Note that the unique solvability for $v$ is similar to that for $v(;t)$ in Step 1, so we omit the proof.  In \eqref{probt}, as before, we split
\eqal{
a_s(;t)=a_s(;0)+r_2(t),\quad a_{p_j}(;t)=a_{p_j}(;0)+r_3(t),\quad a(;t)=a(;0)+r_4(t),
}
where each $r_i(t)$ vanishes in $C^{1,\beta}(\Omega)$ as $t\to 0$.  Thus, omitting $t=0$ arguments, $v(;t)$ solves the inhomogeneous equation
\eqal{
\pd_i(a_sv(;t)u_i+a_{p_j}v_j(;t)u_i+av_i(;t))=o(1)_{C^{\beta}(\Omega)},
}
where $o(1)_{C^\beta(\Omega)}$ indicates a quantity vanishing as $t\to 0$ in $C^\beta(\Omega)$, and its uniformity in $t$ with respect to terms including $v$ or $v_i$ follows from Step 1.  We then put $w(;t)=v(;t)-v$ and see that $w(;t)$ uniquely solves the Dirichlet problem
\eqal{
\pd_i(a_sw(;t)u_i+a_{p_j}w(;t)_ju_i+aw(;t)_i)=o(1)_{C^{\beta}(\Omega)},\\
w(;t)|_{\pd\Omega}=0.
}
The right hand side clearly vanishes in $L^p$ for any $p>1$, so it follows from the strong $W^{2,p}$ estimate \cite[Theorem 9.13]{GT} that $\|w(;t)\|_{W^{2,p}(\Omega)}\to 0$ as $t\to 0$.  By Sobolev embedding, we deduce that $v(;t)\to v$ in $C^1$, as desired.
\end{proof}

We next linearize around solutions $u$ with large gradients if we assume that $a(s,p)$ satisfies additional conditions.

\begin{prop}
Let $a$ satisfy uniform lower bound \eqref{uni} and  decay condition \eqref{decay}.  Then for any solution $u$ to \eqref{eq} with smooth boundary data $f$, the linearization of \eqref{eq} exists at $u$.
\end{prop}

\begin{proof}
The proof of Step 2 in Proposition \ref{prop:lin_small} is unchanged, and we repeat Step 1 until \eqref{aijbound}.  To proceed further, we invoke additional conditions \eqref{uni} and \eqref{decay}.  We suppose $t$ is sufficiently small so that
\eqal{
|a_s(;t)Du(;0)|\le\ep_0+o(1)_{C^{1,\beta}}\|Du(;0)\|_{L^\infty}\le 2\ep_0.
}
Applying the Cauchy-Schwartz inequality and the Poincar\'e inequality, we obtain
\eqal{
\int_\Omega|Dw|^2\,dx\le \frac{4\ep_0^2}{\lambda_0}\cdot C(\Omega)\int_\Omega |Dw|^2\,dx.
}
Choosing $\ep_0$ small depending on $\lambda_0,\Omega$ yields the desired conclusion.  The rest of the proof proceeds identically.
\end{proof}

\subsection{Solutions with prescribed boundary jets}
\label{sec:jet}

\medskip
We first introduce a coordinate normalization.
\begin{defn}\label{Omega}
We say that \textit{$\Omega$ sits above the origin}, if $\{x^n=0\}$ is a supporting hyperplane for $\Omega$ with $0\in\pd\Omega$,
and $\Omega\subset\{x^n>0\}$.
\end{defn}

The class of PDEs \eqref{eq} with conditions \eqref{aij} and \eqref{grow} considered is invariant under Euclidean isometries.  Let $I=R\circ T$ be a Euclidean isometry (rotation $R$ plus translation $T$) such that $I^{-1}(\Omega)$ sits above the origin.  The resulting PDE for $I^*u(x):=u(Ix)$ is also of class \eqref{eq}:
\eqal{\label{eqR}
\nabla\cdot \left[R_*a(I^*u,\nabla I^*u)\nabla I^*u\right]=0,\qquad x\in I^{-1}(\Omega),
}
where $R_*a(s,p)=a(s,R^{-1}p)$.  It is clear that $R_*a$ satisfies any of conditions \eqref{aij}, \eqref{grow}, or \eqref{decay} if and only if $a$ does.

\medskip
We now construct solutions with prescribed boundary jet, provided the gradient at the boundary point is sufficiently small compared to the solution.  No additional assumptions on $a$ are needed.  Recall $\Pi_1(s)$ as in Proposition \ref{prop:lin_small}.

\begin{prop}
\label{prop:open}
Let $\Pi:\re\to(0,\infty)$ be the positive continuous function defined in \eqref{Pidef}.  Then for any $x_0\in\pd\Omega$ and $(s,p)\in\re\times\re^n$ such that $|p|<\Pi(s)$, there exists boundary data $f\in C^\infty(\pd\Omega)$ for \eqref{eq} such that $u(x_0)=s$ and $\nabla u(x_0)=p$, with $\|f-s\|_{C^{2,\al}}\le \Pi_1(s)$
\end{prop}

\begin{proof}

\textbf{Step 1.} We construct sub/super-solutions.

\medskip
First suppose $\Omega$ sits above the origin and $x_0=0$.  Given $(s,p)\in\re\times\re^n$ with $p=p'+p_ne_n$, we define
\eqal{
u^{s,p}(x):=s+p'\cdot x-Ap_n\log(1-x^n/A),
}
where $A$ is the sufficiently large constant
\eqal{\label{Adef}
A:=2\text{ diam }\Omega\ge 2\max_{x\in\Omega}x^n.
}
We have $u^{s,p}(0)=s$ and $Du^{s,p}(0)=p$.  The derivatives:
\eqal{
Du^{s,p}(x)=p'+\frac{p_n}{A-x^n}e_n,\quad D^2u^{s,p}(x)=\frac{p_n}{(A-x^n)^2}e_n\otimes e_n.
}
The expanded form of equation \eqref{eq} is
\eqal{\label{exp_eq}
a_{ij}u_{ij}+a_s|\nabla u|^2=0.
}
For $\pm 1=\text{sign }p_n$, we thus obtain
\eqal{
\pm (a_{ij}u^{s,p}_{ij}+a_s|Du^{s,p}|^2)&\ge \lambda_0(|u^{s,p}|)\frac{|p_n|}{(A-x^n)^2}-\mu_0(|u^{s,p}|)(|p'|^2+\frac{p_n^2}{(A-x^n)^2})\\
&\ge \frac{\mu_0(U')}{(A-x^n)^2}\left[\frac{\lambda_0(U')}{\mu_0(U')}|p_n|-(\frac{A^2}{4}|p'|^2+p_n^2)\right],
}
where
$$
U':=s+\frac{A}{2}\,|p'|+A|p_n|\log 2
$$
satisfies $|u^{s,p}(x)|\le U'$ on $\Omega$.

\medskip
Let us now restrict $p$ to satisfy
\eqal{\label{Udef}
\max(|p'|,|p_n|)\le 1,\qquad U'\le U:=s+2A.
}
Then we obtain
\eqal{
\pm (a_{ij}u^{s,p}_{ij}+a_s|Du^{s,p}|^2)&\ge\frac{\mu_0(U)}{(A-x^n)^2}\left[\frac{\lambda_0(U)}{\mu_0(U)}|p_n|-(\frac{A^2}{4}|p'|^2+p_n^2)\right].
}
Choosing now $p$ to be any vector defined in the paraboloid
\eqal{
\label{para0}
\left(\frac{2\lambda_0(U)}{A^2\mu_0(U)}\right)^{-1}|p'|^2\le|p_n|\le \frac{\lambda_0(U)}{2\mu_0(U)},\qquad \max(|p'|,|p_n|)\le 1,
}
we find that $u^{s,p}$ is a one-sided solution:
\eqal{
\pm (a_{ij}u^{s,p}_{ij}+a_s|Du^{s,p}|^2)&\ge 0.
}
To remove the ``corners" of the paraboloid \eqref{para0}, we denote by $U=U^s$ the constant in \eqref{Udef}, and denote the various constants by
$$
C^1_s:=\min(\frac{\lambda_0(U^{s})}{2\mu_0(U^{s})},1),\qquad C^2_s:=\min(\frac{2\lambda_0(U^{s})}{A^2\mu_0(U^{s})},1),
$$
so that after shrinking the bounds, the paraboloid becomes
\eqal{
\label{para1}
\frac{1}{C^2_s}|p'|^2\le |p_n|\le C^1_s.
}
To satisfy the smallness condition of Proposition \ref{prop:lin_small}, we shrink the bounds again.  Defining
$$
B^1_s:=\min(C^1_s,\frac{\Pi_1(s)}{N\text{ diam }\Omega}),\qquad B^2_s:=C^2_s,
$$
for some sufficiently large $N$, it is easy to verify that $u^{s,p}$ satisfies the desired estimate
$$
\|u^{s,p}-s\|_{C^{2,\al}(\pd\Omega)}\le \Pi_1(s)
$$
provided $p=(p',p_n)$ lies within the paraboloid
\eqal{
\label{para}
\frac{1}{B^2_s}|p'|^2\le |p_n|\le B^1_s.
}
We next construct solutions with boundary jet inside this paraboloid, and more generally in the smallest rectangle containing it.

\medskip
\textbf{Step 2.} A comparison principle argument.

\medskip
Given $s$, we let $p\in \re^{n-1}$ be any vector in the vertical boundary of the paraboloid \eqref{para}:
\eqal{\label{parabdy}
|p'|^2<B^1_sB^2_s,\qquad p_n=B^1_s.
}
Since $p_n>0$, it follows that $u^{s,p}$ is a subsolution of \eqref{eq}.  Choosing boundary data $f\in C^\infty(\pd\Omega)$ of the form
\eqal{
f^{s,p}(x):=u^{s,p}(x),\qquad x\in\pd\Omega,
}
the comparison principle \cite[Theorem 10.7]{GT} shows that
\eqal{
u[f^{s,p}](x)\ge u^{s,p}(x),\qquad x\in\Omega,
}
with equality at $x=0$, so the inner normal derivatives satisfy the same inequality:
\eqal{
\frac{\pd u[f^{s,p}]}{\pd x^n}(0)\ge \frac{\pd u^{s,p}}{\pd x^n}(0)=p_n.
}
If we instead choose
$$
p_n=-B^1_s,
$$
then the inequalities reverse because $u^{s,p}$ becomes a supersolution.  This means that the set of normal derivatives with tangential jet $(s,p')$ and \textit{small} boundary data,
\eqal{
\mc N^{s,p'}:=\{\frac{\pd u[g]}{\pd x^n}(0)\Big| g\in C^\infty(\pd\Omega),g(0)=s,\nabla g(0)=p', \text{ and }\\
\|g-s\|_{C^{2,\al}(\pd\Omega)}\le \Pi_1(s)\},
}
contains numbers at least as large as $B^1_s$ and at least as small as $-B^1_s$.  Since the set of $g$ with boundary jet $(g,\nabla g)(0)=(s,p')$ is convex, it is connected.  So, of course, is the set of such $g$ also satisfying the smallness estimate
\eqal{
\label{gsmall}
\|g-s\|_{C^{2,\al}(\pd\Omega)}\le \Pi_1(s).
}
By the smoothness of solutions to \eqref{eq}, we deduce that $\mc N^{s,p'}$ is connected, and it therefore contains the interval $[-B^1_s,B^1_s]$.  Varying $p'$ according to \eqref{parabdy}, we conclude that for each $s$ and any vector $p$ in the rectangle
$$
|p'|^2\le B^1_sB^2_s,\qquad |p_n|\le B^1_s,
$$
there exists a solution with boundary jet $(u(0),\nabla u(0))=(s,p)$ with boundary data $g$ satisfying \eqref{gsmall}.  Defining $\Pi$ by the radius of the largest sphere inside the rectangle,
\eqal{\label{Pidef}
\Pi(s)=\min(\sqrt{B^1_sB^2_s},B^1_s).
}
This completes the proof of the Proposition, in the case that $\Omega$ sits above the origin.

\medskip
\textbf{Step 3.} Rotational invariance.

\medskip
Now suppose $\Omega\subset\re^n$ is any smooth domain, and choose $s,p$ such that $|p|<\Pi(s)$, and $x_0\in\pd\Omega$.  Letting $T(x)=x-x_0$ translate $x_0$ to the origin, we choose a rotation $R\in O(n)$ such that the inner normal vector is mapped to $e_n$, i.e. $R(-\nu|_{x_0})=e_n$.  Then for Euclidean isometry $I=R\circ T$, we see that $I^{-1}(\Omega)$ sits above the origin.  Moreover, inequalities \eqref{aij}, \eqref{coer}, and \eqref{grow} and the constant $A$ in \eqref{Adef} are preserved under the isometry.  Since $|Rp|<\Pi(s)$, we can find a solution $I^*u$ of the rotated equation \eqref{eqR} such that $I^*u(0)=s$ and $\nabla I^*u(0)=Rp$.  In original variables, we have $u(x_0)=s$ and $\nabla u(x_0)=p$, which completes the proof.
\end{proof}

If we also assume the decay condition \eqref{decay}, then we can choose $\Pi=+\infty$.  That is, we can find a solution with any prescribed boundary jet.  We can also weaken conditions \eqref{uni} and \eqref{decay}, in this proof, to
\eqal{
\label{decay1}
|a_s(s,p)|\le\,C\,\frac{\lambda(s,p)}{|p|},
}
where $\lambda(s,p)$ is in \eqref{aij}, and $C>0$ is arbitrary.  This condition makes \eqref{eq} behave similar to a linear equation which admits exponential solutions.
\begin{prop}
\label{prop:decay}
Let $a$ satisfy \eqref{decay1}, or in particular \eqref{uni} and \eqref{decay}.  For any $(s,p)\in\re\times\re^n$ and $x_0\in\pd\Omega$, there exists boundary data $f\in C^\infty(\pd\Omega)$ for \eqref{eq} such that $u(x_0)=s$ and $\nabla u(x_0)=p$.
\end{prop}

\begin{proof}
As in Step 3 in the proof of Proposition \ref{prop:open}, by the invariance of the problem under Euclidean isometries, it suffices to assume that $\Omega$ sits above the origin.

\medskip
Fix $(s,p)\in\re\times \re^{n}$ with $p_n\neq 0$ and $p':=p-p_ne_n$, and for $h>0$ small depending on the constant $C$ in \eqref{decay1} and $p$, we introduce the function
\eqal{
u^{s,p}(x):=s+p'\cdot x+h\,p_n\,(e^{x^n/h}-1).
}
If $p_n>0$ (resp. $p_n<0$) then we claim that $u^{s,p}$ is a subsolution (supersolution) of \eqref{eq}, with jet
\eqal{
u^{s,p}(0)=s,\qquad \nabla u^{s,p} (0)=p.
}
We have
\eqal{
|\nabla u^{s,p}|^2=|p'|^2+p_n^2e^{2x^n/h},\qquad u^{s,p}_{ij}=\frac{p_n}{h}e^{x^n/h}\delta_{in}\delta_{jn},
}
so ellipticity \eqref{aij} yields, for $\pm 1=\text{sign } p_n$,
\eqal{
\pm(a_s|\nabla u^{s,p}|^2+a_{ij}u^{s,p}_{ij})\ge \pm a_s|\nabla u^{s,p}|^2+\lambda \frac{|p_n|}{h}e^{x^n/h}.
}
Applying decay condition \eqref{decay1}, we obtain
\eqal{
\pm(a_s|\nabla u^{s,p}|^2+a_{ij}u^{s,p}_{ij})\ge \lambda e^{x^n/h}\left(\frac{|p_n|}{h}-C\sqrt{|p'|^2e^{-2x^n/h}+p_n^2}\right).
}
Choosing $h$ such that $|p_n|/h\ge C\sqrt{|p'|^2+p_n^2}$ verifies the claim.

\medskip
We next observe that by continuity, the set of possible normal derivatives with tangential jet $(s,p')$
\eqal{
\{\frac{\pd u[g]}{\pd x^n}(0)|\quad g\in C^\infty(\pd\Omega) \text{ and }g(0)=s,\nabla'g(0)=p'\}
}
is connected and nonempty, so to show it is $\re$, we need only show it is unbounded from above and below.  If $p_n\gg 1$ is large, we let $f=u^{s,p}|_{\pd\Omega}$ and $u$ solve \eqref{eq} with boundary value $f$.  Then $u^{s,p}$ is a subsolution, and the comparison principle \cite[Theorem 10.7]{GT} says
$$
u(x)\ge u^{s,p}(x),\qquad x\in\Omega,
$$
with equality at $x=0$, so
$$
\frac{\pd u}{\pd x^n}(0)\ge \frac{\pd u^{s,p}}{\pd x^n}(0)=p_n.
$$
Repeating the argument with $p_n\ll -1$ and using supersolution $u^{s,p}$ yield the desired conclusion.
\end{proof}

\begin{rem}
The results of this section hold true unchanged for non-homogeneous conductivities $a=a(x,s,p)$, provided the inequalities \eqref{aij} and \eqref{grow} remain independent of $x$.  We instead define the transformed conductivity in \eqref{eqR} by $I_*a(x,s,p)=a(Ix,s,R^{-1}p)$.
\end{rem}

\subsection{Geometric reformulation of the linearized problem}
\label{sec:geo}
We next geometrically reformulate the linearized equation \eqref{eqL} and its DN map \eqref{DNL}, see e.g. \cite{LU} for the same with the conductivity equation via the Laplace operator.  We assume throughout this section that the linearization of \eqref{eq} exists at a given solution $u$, which we addressed in Section \ref{sec:lin}

\medskip
Let us first rewrite the equation.  We can decompose the principal part of \eqref{eqL}, namely
$$
\pd_i(a_{p_j}v_ju_i+av_i),
$$
into its symmetric and anti-symmetric parts and rewrite the linearized differential operator as:
\eqal{\label{eqL1}
\mc L_a[u]v&:=\pd_i(a_svu_i+a_{p_j}v_ju_i+av_i)\\
&=\pd_i(a_{ij}v_j+b^iv),
}
where linearized conductivity $a_{ij}$, given in \eqref{aij}, is the symmetric part, and
\eqal{\label{bA}
b^i:=a_su_i-\pd_jA_{ij},\qquad A_{ij}:=\frac{1}{2}(a_{p_j}u_i-a_{p_i}u_j)
}
accounts for the anti-symmetric part.  We can also rewrite the linearized DN map \eqref{DNL}:
\eqal{\label{DNL1}
\Gamma_a[u](h)=(a_{ij}\nu_iv_j+A_{ij}\nu_iv_j+a_sv\,u_in_i)dS.
}
Henceforth in this section only, we distinguish Euclidean coordinates from ordinary subscripts (which will denote local coordinates) unless otherwise stated.

\medskip
We next rewrite this equation in terms of geometric quantities.  Let $x^i$ denote local coordinates on $\Omega$, and denote $x^i_e$ to be global Euclidean coordinates.  We define a $(2,0)$ tensor field $G\in \mc T^{2}_0(\Omega)$ by its values in Euclidean coordinates, using the symmetric part $a_{ij}$ in \eqref{aij} of the linearized operator.  If $n\ge 3$,
\eqal{
G^{ij}_e:=G(dx^i_e,dx^j_e)=
(\det a_{k\ell})^{\frac{1}{2-n}}a_{ij},\qquad G=G^{ij}\pd_i\pd_j.
}
If $n=2$, we first define a conductivity function $\sigma\in C^\infty(\Omega)$ by
\eqal{\label{sigma}
\sigma:=\det a_{k\ell}.
}
We can then define $G$ as follows:
\eqal{
G^{ij}_e:=G(dx^i_e,dx^j_e)= (\det a_{k\ell})^{-1/2}a_{ij},\qquad G=G^{ij}\pd_i\pd_j.
}
In each case, the relationship can be written as
\eqal{\label{gaij}
&n\ge 3:&&\frac{1}{\sqrt{\det G^{k\ell}_e}}G^{ij}_e=a_{ij},\\
&n=2:&&\sigma\,G^{ij}_e=a_{ij},\qquad \det G^{k\ell}_e=1.
}
We next define a Riemannian metric $g\in \mc T^0_2(\Omega)$ by the inverse of $G$ in Euclidean coordinates, which is well defined by ellipticity \eqref{aij}
\eqal{\label{g}
g_{e,ij}:=g(\frac{\pd}{\pd x^i_e},\frac{\pd}{\pd x^j_e})=(G_e^{-1})_{ij},\qquad g=g_{ij}dx^idx^j.
}
We will instead denote $\pd_{e,i}=\pd/\pd x^i_e$.  The divergence of the linearized conductivity takes two forms in various dimensions:
\eqal{\label{divg}
&n\ge 3:&& \pd_{e,i}(a_{ij}\pd_{e,j}v)=\sqrt{g_e}\,\Delta_{g}v,\\
&n=2:&&\pd_{e,i}(a_{ij}\pd_{e,j}v)=\text{div}_g(\sigma\nabla_gv)=\sigma\Delta_gv+\langle \nabla_g\sigma,\nabla_gv\rangle_g,
}
where, in local coordinates, the Laplace operator, gradient, and divergence are given by
\eqal{
\Delta_gv=\frac{1}{\sqrt g}\pd_i(\sqrt g\,g^{ij}\pd_jv),\quad \nabla_gv=g^{ij}\pd_jv\,\pd_i,\quad\text{div}_gA=\frac{1}{\sqrt g}\pd_i(\sqrt g\,A^i).
}
To account for the lower order terms in \eqref{eqL1}, we define the \textit{magnetic Schr\"odinger operator with potential}:
\eqal{\label{mag}
\Delta_{g,A,q}v&:=\frac{1}{\sqrt g}(\pd_i+A_i)\sqrt g\,g^{ij}(\pd_j+A_j)v+qv\\
&=\Delta_gv+2\langle A^\#,\nabla_gv\rangle+\left(
\text{div}_gA^\#+|A|_g^2+q\right)v,
}
where $A=A_idx^i,q$ are to be chosen, $A^\#=g^{ij}A_jdx^i$, and $|A|_g^2=g^{ij}A_iA_j$.  Recalling \eqref{divg}, we obtain the following relations:
\eqal{
\label{eqLgeo}
&n\ge 3:&&\mc L_a[u]v=\sqrt{g_e}\,\Delta_{g,A,q}v,\\
&n=2:&&\mc L_a[u]v=\sigma\,\Delta_{g,A,q}v,
}
provided the following relations are true:
\eqal{\label{Aq}
&n\ge 3:&&b^i=2\sqrt{g_e}\,g^{ij}_eA_{e,j},\qquad \pd_{e,i}b^i=\sqrt{g_e}(\text{div}_gA^\#+|A|_g^2+q),\\
&n=2:&&b^i+g^{ij}_e\pd_{e,j}\sigma=2\sigma g^{ij}_eA_{e,j},\qquad \pd_{e,i}b^i=\sigma(\text{div}_gA^\#+|A|_g^2+q).
}
This system can be solved for $A$ and $q$ in terms of $b,\sigma,$ and $g$.  This completes the reformulation of the linearized PDE \eqref{eqL} via \eqref{eqLgeo}.

\medskip
We next reformulate the linearized DN map \eqref{DNL1}, and start with the anti-symmetric part of the linearized operator \eqref{bA}.  We define a $(1,1)$ tensor $\al\in \mc T^1_1(\Omega)$ in terms of $g$:
\eqal{\label{alA}
\al_{e,j}^i:=\al(dx^i_e,\pd_{e,j})= \frac{1}{\sqrt{g_e}}\,g_{e,jk}A_{ik
,\qquad \al=\al^i_j\pd_idx^j.
}
Here, $\sqrt{g_e}=\sqrt{\det g_{e,ij}}$, and $A_{ik}$ is as in \eqref{bA}.  The tensor $\al$ is anti-symmetric in the inner product induced by $g$:
\eqal{\label{anti}
\langle \al\cdot V,W\rangle_g=-\langle V,\al\cdot W\rangle_g,\quad V,W\in T\Omega,
}
since in Euclidean coordinates, with $V=V^i_e\pd_{e,i},W=W^i_e\pd_{e,i}$, we have
\eqal{
\langle \al\cdot V,W\rangle_g&=g_{e,i\ell}\left(\frac{1}{2\sqrt{g_e}}g_{e,jk}(a_{p_k}u_i-a_{p_i}u_k)V^j_e\right)W^\ell_e\\
&=:(g_{e,i\ell}T^{ik}g_{e,jk})V^j_eW^\ell_e,
}
where $T^{ik}$ is anti-symmetric in its indices, so the claim follows.

\medskip
The Riemannian volume form $dV_g$ has the Euclidean representation
\eqal{
dV_g=\sqrt{\det g_e}\,dx^1_e\wedge\cdots\wedge dx^n_e=\sqrt{\det g_e}\,dx.
}
The induced area form $dS_g$ on $\pd \Omega$ is
\eqal{
dS_g=\nu_g\,\lrcorner\, dV_g|_{\pd\Omega},
}
where $\nu_g$ is the unit normal to $\pd\Omega$.  Letting $f$ be a defining function for $\pd\Omega$, the unit normals for Euclidean and $g$ metrics can be written in Euclidean coordinates as
\eqal{
\nu=(|df|_e)^{-1}\pd_{e,i}f\pd_{e,i},\qquad \nu_g=(G(df,df))^{-1/2}G^{ij}_e\pd_{e,j}f\pd_{e,i}.
}
We have the following relationship between the normal vectors.  This allows us to reformulate the part of the linearized DN map with anti-symmetric part $A_{ij}$.  This is likely well known, but we give the proof for completeness.

\begin{lem}\label{lem:normal}
As sections of the product bundle
$
\pd(T\Omega)\otimes \Lambda^0_{n-1}(\pd\Omega)$, the following equality holds:
\eqal{\label{product}
&n\ge 3:&&(a_{ij}\nu_j\pd_i)\otimes dS=\nu_g\otimes dS_g,\\
&n=2:&& (a_{ij}\nu_j\pd_i)\otimes dS=\sigma\,\nu_g\otimes dS_g.
}
In particular,
\eqal{\label{product_cases}
&n\ge 3:&&(a_{ij}\nu_iv_j+A_{ij}\nu_iv_j)dS=\langle \nabla_g v+\al\cdot\nabla_gv,\nu_g\rangle_g dS_g,\\
&n=2:&& (a_{ij}\nu_iv_j+A_{ij}\nu_iv_j)dS=\langle \sigma\,\nabla_g v+\al\cdot\nabla_gv,\nu_g\rangle_g dS_g,
}
where $v\in C^\infty(\ov\Omega)$, $a_{ij},\sigma$ are in \eqref{gaij}, $A_{ij}$ is in \eqref{bA}, and $\al$ is in \eqref{alA}.
The local coordinates used in this lemma and proof are Euclidean, without subscript $e$.
\end{lem}

\begin{proof}
We will show the identities at each $x_0\in\pd\Omega$.  The setting of the lemma is unchanged after applying a Euclidean isometry because the class of PDEs \eqref{eq} is unchanged by such.  After a Euclidean isometry, we may assume that $x_0=0$ is the origin, and that $\Omega$ sits above the origin, recall Definition \ref{Omega}.  The tangent space $T_0(\pd\Omega)$ at the origin is the span of $\pd_1,\dots,\pd_{n-1}$, and $f=x^n$ is a defining function for $\pd\Omega$ at $x=0$, so the unit normal vectors are
\eqal{
\nu|_{x=0} = \pd_n,\qquad \nu_g|_{x=0}=(G_{nn})^{-1/2}G^{in}\pd_i.
}
Since $\pd_i\,\lrcorner \,dV_g|_{x^n=0}=0$ for $i<n$, this means the area forms at $x=0$ are
\eqal{
dS|_{x=0}=dx^1\wedge\cdots\wedge dx^{n-1},\qquad dS_g|_{x=0}=\sqrt{G_{nn}}\sqrt{g}\,dx^1\wedge\cdots \wedge dx^{n-1}.
}
We find that
\eqal{
\nu_g\otimes dS_g|_{x=0}=(\sqrt{g}\,G^{in}\pd_i)\otimes dS,
}
while \eqref{gaij} yields
\eqal{
&n=3:&& (a_{ij}\nu_j\pd_i)\otimes dS|_{x=0}=(\sqrt{g}\,G^{in}\pd_i)\otimes dS,\\
&n=2:&& (a_{ij}\nu_j\pd_i)\otimes dS|_{x=0}=(\sigma\, G^{in}\pd_i)\otimes dS,
}
with $\det {g}=1$ if $n=2$, which verifies \eqref{product}.

\medskip
That the $a_{ij},\nabla_gv$ terms in \eqref{product_cases} are equal follows from left-contracting \eqref{product} against the $(0,1)$ tensor $dv$.  If we instead contract against $(\al\cdot\nabla_gv)_\flat$, where $A_\flat=g_{ij}A^jdx^i$, and if we recall \eqref{alA}, then we obtain for $n=2$,
\eqal{\label{Aij_calc}
\langle \al\cdot\nabla_gv,\nu_g\rangle dS_g&=\frac{1}{\sigma}\,g_{ik}(\al^k_\ell g^{\ell m}v_m)a_{ij}\nu_jdS\\
&=g_{ik}(g_{\ell s}A_{ks}g^{\ell m}v_m)g^{ij}\nu_jdS\\
&=A_{js}v_s\nu_j dS.
}
The $n\ge 3$ case is similar.  This completes the proof.
\end{proof}

For the final term in \eqref{DNL1}, we define a vector field $\beta=\beta^i\pd_i$ so that the following equation holds on $\pd\Omega$:
\eqal{\label{beta}
&n\ge 3:&& (a_su_in_i)dS=\langle A^\#+\beta,\nu_g\rangle dS_g,\\
&n=2:&&(a_su_in_i)dS=\langle \sigma A^\#+\beta,\nu_g\rangle dS_g.
}
We now recall the \textit{magnetic Dirichlet-to-Neumann map}, $C^\infty(\pd\Omega)\to C^\infty(\pd\Omega):$
\eqal{\label{Lambda}
&\Lambda_{g,A,q}(h):&&h\mapsto \langle \nabla_gv[h]+A^\#v[h],\nu_g\rangle,\\
&\text{where:}&&\Delta_{g,A,q}v=0,\\
& &&v|_{\pd\Omega}=h.
}
Then we have the following geometric reformulation.  The linearized PDE \eqref{eqL1} corresponds to a magnetic Schr\"odinger operator with potential, and the linearized DN map \eqref{DNL1} corresponds to an oblique magnetic normal derivative with lower order term.

\begin{prop}\label{prop:geo}
Let $u$ be a solution of \eqref{eq} and $f\in C^\infty(\pd\Omega)$ such that the linearization of \eqref{eq} exists at $u$.  Let also $h\in C^\infty(\pd\Omega)$ and $v$ solve the linearized problem \eqref{eqL}.

\medskip
\noindent
Then for $n\ge 3:$
\eqal{\label{n3}
\mc L_a[u]v&=\sqrt{g_e}\,\Delta_{g,A,q}v,\\
\Gamma_a[u](h)&=\Lambda_{g,A,q}(h)dS_g+\langle\al\cdot\nabla_gh+\beta h,\nu_g\rangle_g dS_g.
}
For $n=2$:
\eqal{\label{n2}
\mc L_a[u]v&=\sigma\,\Delta_{g,A,q}v,\\
\Gamma_a[u](h)&=\sigma\Lambda_{g,A,q}(h)dS_g+\langle\al\cdot\nabla_gh+\beta h,\nu_g\rangle_g dS_g.
}
Here, metric $g$ is given by \eqref{gaij},\eqref{g}, and $\sigma$ by \eqref{sigma} for $n=2$.  Magnetic potential $A$ and potential $q$ are determined by \eqref{Aq}, anti-symmetric tensor $\al$ is given by \eqref{alA}, and vector field $\beta$ is defined according to \eqref{beta}.  The map $\Lambda$ is defined by \eqref{Lambda}, and $\Gamma_a[u]$ is defined by \eqref{DNL1}.

\medskip
\noindent
In particular, if $\Gamma_a=\Gamma_{\ti a}$, then for $n\ge 3$:
\eqal{\label{DNn3}
\Lambda_{g,A,q}(h)dS_g+\langle\al\cdot\nabla_gh+\beta h,\nu_g\rangle_g dS_g=\\
\Lambda_{\ti g,\ti A,\ti q}(h)dS_{\ti g}+\langle\ti \al\cdot\nabla_{\ti g}h+\ti \beta h,\nu_{\ti g}\rangle_{\ti g} dS_{\ti g}.
}
For $n=2:$
\eqal{\label{DNn2}
\sigma\Lambda_{g,A,q}(h)dS_g+\langle\al\cdot\nabla_gh+\beta h,\nu_g\rangle_g dS_g=\\
\ti\sigma \Lambda_{\ti g,\ti A,\ti q}(h)dS_{\ti g}+\langle\ti \al\cdot\nabla_{\ti g}h+\ti \beta h,\nu_{\ti g}\rangle_{\ti g} dS_{\ti g}.
}
\end{prop}

\begin{proof}
The only claim remaining to prove is the identity
\eqal{
\langle \al\cdot\nabla_gv[h],\nu_g\rangle_g dS_g=\langle \al\cdot\nabla_gh,\nu_g\rangle_g dS_g,
}
or that this term depends only on the tangential projection of $\nabla_gv[h]$.  This follows from the anti-symmetry of $\al$, since the normal contribution is
$$
\langle\nabla_gv[h],\nu_g\rangle\langle\al\cdot\nu_g,\nu_g\rangle=0,
$$
which follows from \eqref{anti}.
\end{proof}

\subsection{Boundary determination of the linearized conductivity}
\label{sec:bdy}
In this section, we determine the tangential part of the linearized conductivity $a_{ij}$ at the boundary $\pd\Omega$.  The main technical tool we need is the calculation of the symbol of the magnetic DN map \eqref{Lambda} in boundary normal coordinates, \cite[Lemma 8.6]{DKSU}.  See \cite{LU} for the construction in the case of the ordinary Laplace operator.

\medskip
Letting $x^1,\dots,x^{n-1}$ be local coordinates on a neighborhood $\Sigma\subset\pd\Omega$, we denote boundary normal coordinates in a neighborhood of $\Sigma$ in $\Omega$ by $(x^1,\dots,x^n)$, with $x^n$ the $g$-distance function from $\pd\Omega$.  The metric and its dual in boundary normal coordinates are
\eqal{
g=(dx^n)^2+\sum_{i,j<n}g_{ij}dx^idx^j,\quad G=\pd_n\otimes \pd_n+\sum_{i,j<n}g^{ij}\pd_i\otimes \pd_j.
}
Note that
\eqal{
\sum_{i,j<n}g_{ij}dx^idx^j=g|_{T\pd\Omega},\qquad \sum_{i,j<n}g^{ij}\pd_i\otimes \pd_j=G|_{T^*\pd\Omega},
}
if $x\in\pd\Omega$.  For two metrics $g,\tilde g$, we use the \textit{same} local coordinates $x^1,\dots,x^{n-1}$ for $\Sigma$ when defining boundary normal coordinates.

\medskip
We also need the principal symbol of pseudo-differential operators on a manifold using left quantization, see e.g. \cite[Definition 18.1.20]{H}.  The differential operator $C^\infty(\pd\Omega)\to C^\infty(\pd\Omega)$ defined by
$$
h\mapsto\langle\al\cdot\nabla_gh+\beta h,\nu_g\rangle_g=\sum_{k<n}(-\al\cdot\nu_g)^kh_k+\langle\beta,\nu_g\rangle h,
$$
clearly has principal symbol
\eqal{
i\langle\al\cdot\xi^\#,\nu_g\rangle_g=\sum_{k<n}(-\al\cdot\nu_g)^k(i\xi)_k,\qquad (x,\xi)\in T^*_x(\pd\Omega).
}
To compute the final term in the DN maps \eqref{n3},\eqref{n2}, we now apply \cite[Lemma 8.6]{DKSU}, see also \cite{LU}, where the symbol of the magnetic DN map \eqref{Lambda} is computed in boundary normal coordinates in $n\ge 3$ dimensions.  The proof there goes through unchanged for the case of $n=2$, although the subsequent boundary determination problem is different due to conformal invariance; see the remark in \cite{LU}.

\begin{comment}
For completeness, we calculate the principal symbol of the two differential operators in \eqref{n3}

\begin{lem}
Let $\beta$ be a smooth vector field on $\pd\Omega$.  The differential operator
$$
C^\infty(\pd\Omega)\to C^\infty(\pd\Omega;T\pd\Omega):\qquad h\mapsto\nabla_gh+\beta h,
$$
has principal symbol $i\xi^\#=g^{jk}\xi_k\pd_j$, where $(x,\xi)\in T^*_x(\pd\Omega)$.
\end{lem}

\begin{proof}
Let $\Sigma\subset\pd\Omega$ be the neighborhood for some coordinate chart $\Sigma\to\tilde\Sigma\subset\re^{n-1}$, and $\psi\in C^\infty_0(\tilde \Sigma)$.  By the Fourier inversion formula,
$$
\psi(x)=\frac{1}{(2\pi)^{n-1}}\int e^{i\langle x-y,\xi\rangle}\psi(y)dyd\xi,
$$
so applying $\nabla_g+\beta=g^{jk}\pd$ in coordinates yields
$$
\nabla_g\psi+\beta\psi=\frac{1}{(2\pi)^{n-1}}\int e^{i\langle x-y,\xi\rangle}(i\xi^\#+\beta)\psi(y)dyd\xi.
$$
The first order homogeneous part is thus $i\xi^\#$.
\end{proof}
\end{comment}

\begin{lem}[From Lemma 8.6 in \cite{DKSU}]
Let $(\Omega,g)$ be a compact Riemannian manifold with boundary with dimension $n\ge 2$.  Then $\Lambda_{g,A,q}$ is a pseudo-differential operator of order 1 on $\pd \Omega$, and its principal symbol i
\eqal{
-|\xi|_g:=-\sqrt{\sum_{i,j<n}g^{ij}(x)\xi_i\xi_j},\qquad (x,\xi)\in T^*_x\pd \Omega.
}
\end{lem}

\medskip
We thus obtain the principal symbol of $\Gamma_a[u]$, thought of as a form-valued pseudo-differential operator:
\eqal{\label{sym}
&n\ge 3:&&(-|\xi|_g+i\langle\al\cdot\xi^\#,\nu_g\rangle)dS_g,\\
&n=2:&&(-\sigma|\xi|_g+i\langle\al\cdot\xi^\#,\nu_g\rangle)dS_g.
}

In dimension $n\ge 3$, we can recover the tangential projection of the metric on $\pd\Omega$.  In dimension $n=2$, we can recover the ``effective conductivity" $\sigma$ on the boundary.  In both cases, we can recover the normal elements of the anti-symmetric tensor.

\begin{lem}\label{lem:O1}
If $\Gamma_a=\Gamma_{\ti a}$, then $G|_{T^*\pd\Omega}=\ti G|_{T^*\pd\Omega}$ if $n\ge 3$, where $G$ is given in terms of $a$ in \eqref{gaij}.  If $n=2$, then $\sigma=\ti\sigma$ on $\pd\Omega$, where we defined $\sigma$ by \eqref{sigma}.  In both cases, we have
\eqal{\label{alN}
(\al\cdot\nu_g)\,dS_g=(\ti\al\cdot\nu_{\ti g})\,dS_{\ti g}
}
on $\pd\Omega$.
\end{lem}

\begin{proof}
By Proposition \eqref{prop:geo}, equalities \eqref{n3}, \eqref{n2} hold, so the principal symbols on $T^*\pd\Omega$ of the associated pseudo-differential operators on $\pd\Omega$ must coincide.  By \eqref{sym} and \eqref{DNn3}-\eqref{DNn2}, we obtain
\eqal{\label{sym1}
&n\ge 3:&&(-|\xi|_g+i\langle\al\cdot\xi^\#,\nu_g\rangle)dS_g=(-|\xi|_{\ti g}+i\langle\ti \al\cdot\ti\xi^\#,\nu_{\ti g}\rangle)dS_{\ti g},\\
&n=2:&&(-\sigma|\xi|_g+i\langle\al\cdot\xi^\#,\nu_g\rangle)dS_g=(-\ti \sigma|\xi|_{\ti g}+i\langle\ti \al\cdot\ti \xi^\#,\nu_{\ti g}\rangle)dS_{\ti g},
}
where we denote $\ti\xi^\#=\ti g^{ij}\xi_j\pd_i$.  In boundary normal coordinates, we have
$$
dS_g=\pd_n\,\lrcorner\, dV_g|_{x^n=0}=\sqrt{g}\, dx^1\wedge\cdots\wedge dx^{n-1}=\sqrt{g|_{T\pd\Omega}}\,dx^1\wedge\cdots\wedge dx^{n-1},
$$
so equating the real parts in \eqref{sym1} yields
\eqal{\label{real}
&n\ge 3:&&(\det g) g^{ij}=(\det \ti g)\ti g^{ij},\qquad i,j<n,\\
&n=2:&&\sigma=\tilde\sigma.
}
The imaginary parts in \eqref{sym1} yield \eqref{alN} in both cases.

\medskip
Taking the determinant of both sides of \eqref{real} shows that $(\det g)^{n-2}=(\det \ti g)^{n-2}$, hence $g^{ij}=\ti g^{ij}$ for $i,j<n$.  The $n\ge 3$ case thus recovers $G|_{T^*\pd\Omega}$.
\end{proof}

We thus obtain the main result of this section, a boundary determination result for the quasilinear PDE \eqref{eq} in Euclidean coordinates.
\begin{prop}
\label{prop:bdy}
Let $u$ be a solution of \eqref{eq} and $f\in C^\infty(\pd\Omega)$ such that the linearization of \eqref{eq} exists at $u$.   Similarly with $\ti u$, and $\ti a$ replacing $a$.

\medskip
If $n\ge 3$ and $\Gamma_a=\Gamma_{\ti a}$, then the following holds on $\pd\Omega$:
\begin{align}
\label{symeq}
[aI+\frac{1}{2}(\nabla_pa\otimes \nabla f+&\nabla f\otimes\nabla_pa)]\Big|_{T^*\pd\Omega}=
\\
\nonumber
&[\ti aI+\frac{1}{2}(\nabla_p\ti a\otimes \nabla  f+\nabla  f\otimes\nabla_p\ti a)]\Big|_{T^*\pd\Omega}.
\end{align}

If $n\ge 2$ and $\Gamma_a=\Gamma_{\ti a}$, then we have on $\pd\Omega$,
\eqal{
\label{aneq}
(\nabla_pa\otimes\nabla u-\nabla u\otimes\nabla_pa)&\cdot \nu=(\nabla_p\ti a\otimes\nabla \ti u-\nabla \ti u\otimes\nabla_p\ti a)\cdot \nu.
}

\medskip
If $n=2$ and $\Gamma_a=\Gamma_{\ti a}$, then we also have
\eqal{\label{deteq}
\det a_{ij}=\det \ti a_{ij}
}
on $\pd\Omega$, where $a_{ij}$ is in \eqref{aij}.
\end{prop}

\begin{proof}
If $\Gamma_a=\Gamma_{\ti a}$, then Lemma \ref{lem:O1} applies.  Since for $n\ge 3$ we have $G|_{T^*\pd\Omega}=\ti G|_{T^*\pd\Omega}$, the characterization \eqref{gaij} of $a_{ij}$ implies \eqref{symeq}; indeed, the restriction of $\nabla u$ to $T^*\pd\Omega$ is precisely $\nabla f$ by \eqref{eq}.  Recalling the definition \eqref{sigma} of $\sigma$ shows that $\det a_{ij}=\det \ti a_{ij}$ on $\pd\Omega$.

\medskip
For \eqref{alN}, we recall Lemma \ref{lem:normal} and \eqref{Aij_calc}, in the form
\eqal{
-(\al\cdot\nu_g)^sdS_g=(A_{js}\nu_j)dS,
}
and \eqref{alN} yields
\eqal{
A_{ij}\nu_j=\ti A_{ij}\nu_j,
}
which is precisely \eqref{aneq}, recalling the definition \eqref{bA} of $A_{ij}$.
\end{proof}

\section{Proofs of Theorems}
\label{sec:proof}

Suppose first that $\Omega$ sits above the origin, see Definition \ref{Omega}.  Evaluating equalities \eqref{symeq}, \eqref{aneq}, \eqref{deteq} at $x=0$ and identifying $T^*_0\pd\Omega$ with $\re^{n-1}\times\{0\}$, we obtain the following equalities at $x=0$:

\medskip
\noindent
If $n\ge 3:$
\begin{align}
\label{geq1}
a\delta_{ij}+\frac{1}{2}(a_{p_j}f_i+a_{p_i}f_j)=\ti a\delta_{ij}+\frac{1}{2}(\ti a_{p_j}f_i+\ti a_{p_i}f_j),\quad i,j<n.
\end{align}
If $n\ge 2:$
\begin{align}
\label{aleq1}
a_{p_i}u_n-a_{p_n}f_i=\ti a_{p_i}\ti u_n-\ti a_{p_n}f_i,\quad i<n.
\end{align}
If $n=2:$
\eqal{
\label{deteq1}
(a+a_{p_1}f_1)(a+a_{p_2}u_2)-\frac{1}{4}(a_{p_1}u_2&+a_{p_2}f_1)^2=(\ti a+\ti a_{p_1}f_1)(\ti a+\ti a_{p_2}\ti u_2)\\
&-\frac{1}{4}(\ti a_{p_1}\ti u_2+\ti a_{p_2}f_1)^2.
}
Our approach is to use convenient choices of $f$ to deduce $a=\ti a$ along a subspace of $\re\times\re^n$, then use Euclidean isometry invariance to rotate this subspace and deduce the full result.  The homogeneity assumption $\pd a(x,s,p)/\pd x=0$ is used in the last step, but we remark in each case how to weaken it to obtain Corollary \ref{cor:x}.  Theorem \ref{thm:decay} is an immediate consequence of the below proof once we replace $\Pi(s)$ with $+\infty$.

\bigskip
\textbf{The case $n\ge 3$}

\medskip
Fixing $s\in\re$, we choose $p\in\re^{n-1}\times\{0\}$ such that $0<|p|<\Pi(s)$.  Using Proposition \ref{prop:open}, we can find $f\in C^\infty(\pd\Omega)$ such that the solution $u=u[f]$ to \eqref{eq} has the following properties:
\eqal{
f(0)=s,\quad \nabla f(0)=p,\quad \frac{\pd u}{\pd x^n}(0) =0,\quad \|f-s\|_{C^{2,\al}(\pd\Omega)}\le\Pi_1(s).
}
By Proposition \ref{prop:lin_small}, the linearization thus exists at $u=u[f]$, and we can invoke Proposition \ref{prop:bdy}.

\smallskip
Since $\Gamma_a=\Gamma_{\ti a}$, we infer at $x=0$,
\eqal{
\ti a \frac{\pd\ti u}{\pd x^n}(0)=a\frac{\pd u}{\pd x^n}(0)=0,
}
and hence $\ti u_n(0)=0$ as well.  Substituting this into \eqref{geq1} yields
\eqal{
a\delta_{ij}+\frac{1}{2}(a_{p_j}p_i+a_{p_i}p_j)|_{(s,p)}=\ti a\delta_{ij}+\frac{1}{2}(\ti a_{p_j}p_i+\ti a_{p_i}p_j)|_{(s,p)},\quad i,j<n.
}
Let us put $\mc A=a-\ti a$:
\eqal{
\label{geq3}
\mc A\delta_{ij}+\frac{1}{2}(\mc A_{p_j}p_i+\mc A_{p_i}p_j)|_{(s,p)}=0,\quad i,j<n.
}
The spectrum of this matrix, listed in increasing order, is
\eqal{\label{spec}
\mc A+\frac{1}{2}p\cdot\nabla'_p\mc A-\frac{1}{2}|p||\nabla_p'\mc A|,\quad \mc A,\dots,\mc A,\quad \mc A+\frac{1}{2}p\cdot\nabla'_p\mc A+\frac{1}{2}|p||\nabla_p'\mc A|,
}
where $\nabla_p'\mc A=(\mc A_{p_1},\dots,\mc A_{p_{n-1}})$, and middle eigenvalue $\mc A$ is listed $n-3$ times.  From \eqref{geq3}, these eigenvalues are all zero.  Subtracting the largest and smallest eigenvalues yields
\eqal{
|\nabla_p'\mc A(s,p)|=0,\qquad s\in\re,\quad p\in \re^{n-1}\times\{0\}, \quad |p|<\Pi(s)
}
so the smallest eigenvalue yields
\eqal{\label{mcA0}
\mc A(s,p)=0,\qquad s\in\re,\quad p\in \re^{n-1}\times\{0\}, \quad|p|<\Pi(s).
}
Now choose any Euclidean isometry $I=R\circ T$ such that $I^{-1}(\Omega)$ sits above the origin.  Then the above argument applies to the rotated PDE \eqref{eqR}, and we obtain
\eqal{
R_*\mc A(s,p)=0,\qquad s\in\re,\quad p\in \re^{n-1}\times\{0\}, \quad|p|<\Pi(s).
}
The set of such rotations $R$ is all of $O(n)$.  Recalling that
$$
R_*\mc A(s,p)=\mc A(s,R^{-1}p)
$$
and varying $R\in O(n)$, we conclude that
\eqal{
\label{last_step}
\mc A(s,p)=0,\qquad s\in\re,\quad p\in \re^{n}, \quad|p|<\Pi(s).
}
This completes the proof of Theorem \ref{thm:open} if $n\ge 3$.

\begin{rem}
\label{rem:n3}
Theorem \ref{thm:open} still holds if $n\ge 3$ and we replace $a(s,p)$ and $\ti a(s,p)$ by
$$
F(x,s,p,b(s,p)\quad \text{ and }\quad F(x,s,p,\ti b(s,p))
$$
where $t\mapsto F(x,s,p,t)$ is injective for each fixed $(x,s,p)$.  To see this, we merely observe that \eqref{mcA0} and the injectivity of $F$ imply
$$
b(s,p)=\ti b(s,p),\qquad s\in\re,\quad p\in\re^{n-1}\times\{0\},\quad |p|<\Pi(s).
$$
The Euclidean invariance argument completes the proof.
\end{rem}

\begin{rem}
\label{rem:bsym}
Another non-homogeneous extension.  Let $a$ be non-homogeneous with a symmetry in one direction:
$$
a=a(x^1,\dots,x^{n-1},s,p).
$$
If $\Omega$ is convex, then Theorem \ref{thm:open} still holds.  Indeed, instead choosing a general vector  $p\in\re^n$ with $|p|<\Pi(s)$, we simply repeat the proof starting at \eqref{geq3} and deduce
$$
\mc A(0,0,\dots,0,s,p)=0
$$
directly.  Since $\Omega$ is convex, we can place $\Omega$ above a hyperplane at any point $x\in\pd\Omega$, so a Euclidean invariance argument means there holds
$$
\mc A(\pi_{\re^{n-1}\times\{0\}}(x),s,p)=0
$$
for any $x\in\pd\Omega$.  Since $\pd\mc A/\pd x^n=0$, this completes the proof.  It is not clear whether this works for $n=2$ or $\Omega$ non-convex.
\end{rem}

\bigskip
\noindent
\textbf{The case $n=2$}

\medskip
Fixing $s\in\re$, we choose $p\in\re$ such that $0<|p|<\Pi(s)$.  Using Proposition \ref{prop:open}, we can find $f\in C^\infty(\pd\Omega)$ such that the solution $u=u[f]$ to \eqref{eq} has the following properties:
\eqal{
f(0)=s,\quad \frac{\pd f}{\pd x^1}(0)=p,\quad \frac{\pd u}{\pd x^2}(0) =0,\quad \|f-s\|_{C^{2,\al}(\pd\Omega)}\le\Pi_1(s).
}
Since $\Gamma_a=\Gamma_{\ti a}$, we infer at $x=0$,
\eqal{
\ti a \frac{\pd\ti u}{\pd x^2}(0)=a\frac{\pd u}{\pd x^2}(0)=0,
}
and hence $\ti u_2(0)=0$ as well.  Substituting into \eqref{aleq1} yields
\eqal{
pa_{p_2}(s,p,0)=p\ti a_{p_2}(s,p,0).
}
Substituting this and $u_2=\ti u_2=0$ into \eqref{deteq1} yields
\eqal{
(a+pa_{p_1})a|_{(s,p,0)}=(\ti a+p\ti a_{p_1})\ti a|_{(s,p,0)}.
}
Putting
\eqal{
\mc A(s,p):=a(s,p,0)^2-\ti a(s,p,0)^2,
}
we deduce
\eqal{
\mc A+\frac{1}{2}p\mc A_p=0,
}
and hence
\eqal{
\mc A(s,p)=c(s)|p|^{-1/2}.
}
Since $\mc A(s,p)$ is smooth, we conclude $c=0$:
\eqal{\label{mcA0n2}
\mc A(s,p)=0,\qquad s\in\re,\quad |p|<\Pi(s).
}
As in the proof for $n\ge 3$, since $O(2)\cdot e_1=S^1$, where $e_1=(1,0)$ and $O(2)$ is the group of orthogonal transformations on $\re^2$, invoking Euclidean isometries completes the proof.

\begin{rem}
\label{rem:n2}
As in Remark \ref{rem:n3}, Theorem \ref{thm:open} still holds for $n=2$ if we replace $a$ and $\ti a$ by
$$
F(x,s,p,b(s,p))\quad \text{ and }\quad F(x,s,p,\ti b(s,p)),
$$
where $t\mapsto F(x,s,p,t)$ is injective for each fixed $(x,s,p)$.  Indeed, from \eqref{mcA0n2}, we obtain
$$
F(0,s,p,0,b(s,p,0))^2=F(0,s,p,0,\ti b(s,p,0))^2,\qquad s,p\in\re,\quad |p|<\Pi(s).
$$
By \eqref{coer}, we have $F\ge 1$, so the map $t\mapsto F(x,s,p,t)^2$ is also injective for fixed $(x,s,p)$, and we obtain
$$
b(s,p,0)=\ti b(s,p,0),\quad s,p\in\re,\quad |p|<\Pi(s).
$$
As before, the Euclidean invariance argument completes the proof.
\end{rem}

\section*{Acknowledgements}
I would like to thank my advisor Prof. Gunther Uhlmann for suggesting this problem, and Ryan Bushling for discussions.  R. Shankar was partially supported by the National Science Foundation Graduate Research Fellowship Program under grant No. DGE-1762114.


Manchester Ice Hockey Players Raise a Storm!
We are a County controlled (Trafford LEA) co- educational school for pupils in the age range 3 to 11 years.
If you are considering Worthington Primary School for your child, please view our information for new parents here.
Please Note that tonight's Key Stage 2 choir club has been cancelled due to Parents' Evening.


Dewitt Morris the owner of the Big Sandy Lodge approached me a few moths ago about skiing in the Southern Wind Rivers. His lodge sits right at the base of the Southern Wind River Mountains at the Big Sandy Trailhead. I have skied a few springs in the Winds and usually you can not get into those areas without a bit of suffering and long approaches. It is very difficult to ski in the Winds until May or June because the roads to access the trailhead are usually not free of snow until May or June depending on the year. I was psyched when D invited me and a crew of friends to snowmobile into the lodge, stay there and check out skiing from the lodge via snowmobile to Wilderness boundary and then skinning the rest of the way. D is interested in doing a few guided ski trips into the Winds a year and we went on recon to see if it would be a possibility. We met D early Friday morning, loaded up the sleds and started our adventure. I have never driven a snowmobile before so this was a new thing for me. It was pretty entertaining and lot harder then I thought, I named my sled green goblin and we bounded from the beginning. The sled only threw me off a few times when I gave it to much throttle.
Learning how to drive the Green Goblin!
After a few sled malfunctions we arrived at the lodge just under 2 hours, this was so amazing to be this close to Winds in March. The Winds to me are such a special place to ski because I grew up spending my summers in these mountains since I was 5 years old. Being out here all alone with no other skiers or people is a pretty surreal experience in such big mountains. We dropped our stuff and headed to ski a peak I have skied a few times called Peak 11930 this peak sits right next to Independent Peak and Temple peak. We were able to take the sleds to the Wilderness Boundary through Sedgewick Meadows. This was about a 20 min sled approach from the lodge. We then skinned from the sleds toward the peak. In just about 2 hours we were on top of the peak just in time for some good afternoon spring snow. We were able to get great views of other ski lines around us one being Temple peak which is one of my favorite mountains which I have skied from both sides but never skied from the summit. We were back at the lodge in about 4 1/2 hours from when we left to ski. The lodge is very rustic and has 2 fire places and sits protected in the trees on Mud Lake. It made for a great cozy place to crash for a few nights. It was a nice change not having to sleep in a tent and cook outside.
Travers booting up our unnamed couloir on Temple.
Starting the boot up Temple Peak!
The next day we made our way back to Sedgewick meadow with the goal of skiing Temple Peak, this day was going to be a bit longer but what was nice is we had lots of daylight. The temps were cool enough to not have to worry about the snow warming to quickly which is usually the case in May or June. Coming into the Temple Peak basin gave way to lots of different options of couloirs and fun open faces. Temple looked a bit wind scoured but looked manageable. Myself, Travers and Redbeard headed up an unknown couloir on the northwest side of Temple. I had skied the couloir next to this one a few years back but it did not look very filled in this year. We started up the couloir and then reached the summit ridge and summit in a few hours. Temple sits just under 13,000 feet and from the summit it drops strait down a few thousand fee to the valley floor. The views are amazing looking into deep lake area and the Cirque of the towers. You could see so many more fun ski lines from this vantage point. We had some pretty good corn skiing off the summit and then some chalky wind blown snow through the couloir. Overall a really great ski. We watched Brian and Nate ski a fun face into a couloir across the way. The looked like little ants dropping in to a big valley. We all met up in the basin hooting an hollering and stoked with our day. D found himself some great corn off the backside of Peak 11930. A very successful day there was something for everyone. We made our way back to the sleds and back to eat some great elk steaks and watch the sunset over the mountains. Not a bad way to end the day after skiing 4,000 ft!
Redbeard skiing down Temple, amazing views!
Final Day Laturio Peak! Nate coming up the pitch!
Travers making it look fun!


text:
Stefan Kanchev
The Bulgaria-born Stefan Kanchev (1915-2001) was a designer who led the applied graphic design scene of Europe from the 1940s to 1990s, throughout Postmodernism and into contemporary times. His prolific works related to various visual design fields, from posters to stamps and videos. In particular, Kanchev was at the forefront of logo design, having personally designed over 2,000 logos during his illustrious career. Not only is his oeuvre magnificent in terms of quantity, but its exceptional artistic quality and degree of perfection well demonstrates Kanchev's ingenious talent. Although he was one of the most distinguished designers during a period when graphic design flourished, his name remains rather unfamiliar in design history. Kanchev was a designer veiled behind his creations, as more people remember him for his logos than his name. A member of graphic design organizations, including AIGA in the U.S. and ICTA in the U.K., Kanchev was discovered internationally for the first time when he was introduced in the Japanese design magazine Idea, in 1994. Today, Kanchev is regarded one of the most significant logo designers, along with Otl Aicher, Anton Stankowski, and Paul Rand.
Logo Room
This exhibition reveals a master who has remained hidden in the history of graphic design, and reilluminates his creative work. Logo Room exhibits original handmade works that capture the master craftsmanship of a graphic designer, as well as presenting reinterpretations of Kanchev's works through a unique perspective rather than delivering his design work, one to one. The results consist of multi-dimensional mediums, such as installation, video, and printed material. These works shed light on something beyond the practical side of logos, thereby encouraging viewers to experience the design characteristics of the logos through diverse mediums.
During the creative process on Kanchev's work, a large amount of original archival materials with his works were collected, which give the opportunity for their formation in a representative collection. Her first presentation in Bulgaria was at the exhibition curated by the author of the project within Sofia Design Week 2009, which received a great response among the general public and the media and gave rise to talk again about the name of Stefan Kanchev. The debut of the exhibition provoked enthusiasm and aroused self-confidence for the presence of such a high-quality Bulgarian design.
The exhibition is curated around four aspects: logos, biographical data, sketches and drafts, and postage stamps.
Logo Book
This is the first detailed study of the work of Bulgarian graphic designer Stefan Kanchev (1915 – 2001), the artist behind more than 2,000 logos. As an artist in a socialist country, he enjoyed a career that no western designer could even dream of, receiving state commissions with astonishing constancy and frequency. Many large and well-known companies and institutions in Bulgaria adorned themselves with his logos, a large portion of which have withstood the test of time and continue to be used, imprinted on the national consciousness to such a degree that erasing them would be unthinkable. His life's work is documented in this book, insofar as it was possible to find the materials today and gather his numerous sketches and drawings. It illustrates why and how Stefan Kanchev has earned himself the reputation of one of the most notable masters of logotype of our time.
In 2021 we commemorate 20 years of Stefan Kanchev's death and on this occasion the Logo Book was republished in Bulgarian and English with the financial support of the Pulp and Paper Institute, whose trademark is designed by Kanchev himself.
After the huge interest in the only comprehensive study on the work of Stefan Kanchev, collected in Logo Book, we are happy to share that you can now find the new edition in our online and offline channels, as well as in the sales network of our partners.

redpajama_set_name: RedPajamaCommonCrawl


text:
Oil spill off Mauritian coast is killing aquatic life
New Delhi, India Published: Aug 12, 2020, 01.22 PM(IST)
A general view shows the bulk carrier ship MV Wakashio, that ran aground on a reef, at Riviere des Creoles, Mauritius, in this handout image obtained by Reuters on August 11, 2020 Photograph:( Reuters )
Mauritius is home to world-renowned coral reefs. Now, thousands of species run the risk of drowning in a sea of pollution
The oil leakage from a Japanese bulk carrier off the coast of Mauritius is causing the ocean life around the carrier to die.
In Pics: Images show the extent of ecological damage caused by oil spill off Mauritian coast
On Tuesday, volunteers fished out scores of dead eels dripping in oil in a bid to skim the oil off the surface in what are considered the most pristine beaches in the Indian Ocean. An estimated 1,000 tonnes of oil is currently in the ocean, and more expected to flow in as the ship nears a breaking point.
Home to renowned reefs
Mauritius is home to world-renowned coral reefs. Now, thousands of species run the risk of drowning in a sea of pollution.
Also read: Mauritius struggles to preserve ecosystem amid catastrophic oil spill
The pristine turquoise waters of Mauritius have turned murky brown. Thousands of endangered species and spectacular corals are at risk. A Japanese ship owned by the Nagashiki shipping company has been oozing fuel into the island nation's seawater.
The MV Wakashio believed to have been carrying 4,000 tonnes of fuel oil ran aground on a coral reef on July 25 and now a massive round-the-clock clean-up drive is underway.
Creative solutions underway
Mauritians have made floating booms of human hair and leaves to mop up the oil from leaking further.
Some hair salons are even offering discounts to those donating hair. A state of emergency has been declared in Mauritius as fishermen try and salvage their precious waters. For now, close to 500 tonnes of oil has been safely pumped out of the water.
What is of grave concern is that large cracks have appeared in the hull of the cargo ship which has prompted the prime minister to warn that it may break in two.
Mauritius is home to some of the most spectacular coral reefs in the world. It's rich flora and fauna attract tourists from across the globe.
Also read: Japanese ship that caused Mauritius oil spill was approved in annual checks
Tourism is a major contributor to this island economy and added around $1.6 billion to its coffers last year. Authorities are worried that the oil spill near the blue bay marine park may endanger the corals, fish and other marine life and locals worry that this catastrophe could hamper their livelihood.
A spokesperson for the Mitsui Osk Lines, the operator of the ship, has profusely apologised for the spill.
An aerial view of the Mauritius coastline serves as a reminder of the havoc wreaked by mankind upon nature. For now, the entire country's ecosystem hangs in the balance.
US Navy Sonar may be causing whale beachings in Pacific: Study
Palestinian president announces long-awaited elections


This fragrant drawer sachet keeps clothes smelling sweet, and makes for the perfect gift.
Recycle your child's beloved old t-shirt by sewing it into a pillow! The results of activity are unique, decorative, creative, and never wasteful.
Instead of heading to the mall to replace worn-out clothes, hold a crafting session by patching up the rips and tears with some stylishly-shaped patches.
Reusable Christmas gift bags are eco-friendly and pretty. Make a batch of reusable Christmas gift bags for your friends and family this holiday season.
Your child can create a special winter headband with the help of some white felt and a length of silver ribbon.
This beginner's sewing craft turns ordinary gloves into fancy fur fashion statements!
Using only straight-line sewing, your fourth can create a nifty felt pencil case perfect for his busy life.
Press pause on your speedy reader's latest adventure with an activity perfect for book lovers.
Here's a year five magic trick to celebrate St. Patrick's Day, with a little practise in using the literary device of alliteration, too.
Give your fourth grader a lesson in making blankets, and he'll be able to give the gift of warmth. Fleece blankets are especially fun to make, and they're cozy!
Invigorate your fourth grader's mind by helping him turn his room into a woven masterpiece using various lengths of material.
Make a poodle skirt for a creative sewing project with your child that will also be a blast from the past!
This zombie shirt is a fun and incredibly simple Halloween costume. Have your kid don the shirt lurch around the neighbourhood searching for brains...or candy!
Teach your fourth grader how to craft her own decorative felt eggs, a perfect way to symbolise spring and new beginnings!
Help your young sewer create a delightful owl egg cosy out of just a few felt scraps and some embroidery floss.


1 - Combining the backlog refinement with the sprint planning saves time during the sprint.
Myth - Sprint planning used to become a really long and boring meeting. Hence teams started practicing Backlog Refinement.
3- Teams can also adjust the size of the current sprint backlog items during the backlog refinement.
Myth - Refinement has nothing to do with this sprint. Its all about planning ahead for the next sprint.
4- Backlog Refinement is a one time event done during the start of the sprint.
Myth - Generally the practice is somewhere mid sprint. But you can do this anytime in the sprint.
5- Backlog refinement is done by the Product Owner and Scrum Master and the team can review it whenever they have time during the sprint.
Myth - Refinement is an activity between the Development team and the Product Owner.
6-The Product owner is optional for refinement, he can send the backlog to be refined through email to the team.
Myth - This is like saying a quarter back is optional for a game of American Football.
7- Backlog refinement can also be used to accept/reject stories of the current sprint.
Myth - Refinement is all about the next Sprint. And has nothing to do with this sprint.
Fact - Thats true. A simple rule is not more than one hour a week of teams time,.
Myth - Done is update may be in Sprint planning . In refinement we do not talk about the "HOW"
10-Bigger items are decomposed into smaller items, items that are no longer valid are deleted from the backlog.
Fact - A part of refinement is figure out how to fit features into a Sprint. This is what is unique in Scrum.
11-Not all team members are mandatory for Backlog Refinement.
Fact , But recommended is the entire team.
True - Grooming and refinement meeting mean the same. These days the Scrum Community tends to call this as refinement .


Many dentists offer a wide range restorative dentistry services, but not all of them have the same commitment to delivering picture-perfect results. At Legacy Dental SA, we don't think our work is done until your smile looks and feel like brand new. That's why we utilize the highest quality, metal-free dental restoration materials. We use tooth-colored composite resins for fillings, and only the highest quality ceramic and zirconia for dental crowns, fixed bridges, and other restoration materials. If you're ready to learn more, call our San Antonio dental office team to learn more or schedule an appointment with us today.
Dental crowns are perhaps the most common metal-free restoration we offer. They sit completely over the top of a damaged or decayed tooth. We utilize dental crowns to replace damaged tooth structure, protect root canal treated teeth, and we can even utilize metal-free dental crowns as part of cosmetic dentistry plans.
The improved appearance of metal-free dental crowns is the main reason why people choose ceramic, porcelain fused to metal, or zirconia dental crowns rather than metal or gold restorations. These quality restorations closely resemble your healthy smile, and they look and function just like your natural smile. In addition to maintaining a natural appearance, ceramics and other metal-free materials are also less likely to cause irritation or sensitivities that can occur with metal restorations.
What is the Metal-Free Restoration Process?
Like dental restorations crafted from other high quality materials, patients need to visit our dental office twice to complete the restoration process. At your first visit, we begin by removing any damaged tooth structure and shaping the tooth in preparation for the placement of your dental crown, bridges, or other restorations. Then, we capture impressions that are used by our dental lab to craft the fixed bridge. Before you leave, we'll place a temporary restoration to protect your prepared teeth. When we receive the final restoration from the lab, you'll return for your second visit. At this shorter appointment, we'll simply remove your temporary and replace it with a custom crown, bridge, or other restoration.
You will still need to care for your smile with regular tooth brushing and flossing, but we may encourage you to switch to softer toothbrushes and less abrasive toothpaste to avoid excessive wear. We may also recommend you use a waxed floss as more fibrous flosses can tug around the edges of your restoration. If you experience nighttime teeth grinding or clenching (bruxism), be sure to wear a mouthguard to protect your restoration. If you participate in athletic events where your smile can come in contact with other players, hard surfaces or sports equipment, you should be sure to wear a mouthguard during these events as well. When possible, avoid bad habits like chewing on ice or other hard objects and don't use your teeth to open packages.


text:
TRAIN SCHEDULES
The North Borneo Railway features a British Vulcan steam locomotive. The engine is designed for wood burning, a costly yet more environmental form of steam. Refurbished to create the nostalgic romance of the bygone era of the Colonial days of British North Borneo, the train allows all passengers to experience the beauty of countryside and the unique sights and sound of Sabah landscape on board a vintage steam engine train.
The North Borneo Railway features a British Vulcan steam locomotive. The engine is designed for wood burning, a costly yet more environmental form of steam. Stepping into the engine is an opportunity to jump back into the shoes of a child, bringing back childhood dreams of pulling on the whistle as the massive train rolls through the countryside. The train also features five Japanese-designed carriages that have been meticulously restored and renovated to reflect the era of the steam train. Although the carriages were built in the 1970s, both the exterior and interior provide an environment that is typical of stepping onto a train in the 1900s. The exterior utilizes the traditional deep green and cream of the original North Borneo Railway, with carved brass logos on the sides measuring 3 feet by 2 feet on each carriage. The interior highlights the natural woods of Sabah. Each set seats four passengers, with a table in between to facilitate the dining experience.
The train runs every Wednesday and Saturday only.
Boarding Time: 9.30am Tanjung Aru Station
Passengers are offered free seating on one of the five colonial-style train carriages.
Each carriage provides the amenity of a modern unisex washroom.
Each carriage accommodates 16 passengers, with a total train capacity of 80 passengers.
Windows remain open throughout the journey to enhance the overall heritage experience.
High-powered fans line the ceilings to ensure maximum comfort.
Early booking is required prior a day before departure.
View Gallery >>
OFFICE OPERATION HOUR
Tel: +6088-448409
Fax: +6088-448509
Email: tickets@northborneorailway.com
Copyright © 2017 Northborneorailway.com. All Rights Reserved


text:
"Reproduced with permission from Nano Letters 2020 20 (3), 1808-1818; copyright American Chemical Society."
The COMPASS project significantly enlarged the nanocrystal material platform and the related applications. From the material side, the synthesis of the more traditional dichalcogenide nanocrystals was significantly advanced by, for example, developing a library of selenourea precursors for their synthesis, and by the demonstration of upscaled fabrication through robotized processes. Our advances on surface functionalization and self-assembly of the particles have led to films with improved properties, such as in the upconversion of the emitted light, in light-emitting diodes (LEDs), solar concentrators, and in biomedical applications. Water-soluble and bio-compatible polymer beads have been developed that can be used as carriers for light emitting, plasmonic, and magnetic nanoparticles, which enabled to combine different functionalities, such as tracking, guiding, and treatment (via hyperthermia) into single object.
COMPASS also contributed strongly to the advancement of colloidal metal-halide perovskite nanocrystals which emerged only recently as a promising material. Here the COMPASS team developed several synthesis protocols and methods for increasing their stability in different environments, including ion-exchange methods, the use of different amines as ligands, 0D to 3D transition mechanisms, two-dimensional layered perovskites, and lead-free double perovskite structures. Applications of these materials in LEDs and lasing structures, solar cells, photodetectors, and in catalysis could be demonstrated as a proof of concept. The photophysics of these materials was studied in great depth, and the gained knowledge led to the design of novel photonic and plasmonic systems that could be employed to boost their performance in optical applications.
So far, the COMPASS project resulted in 58 high-impact publications and several patent applications.
More than forty researchers were involved in the scientific exchanges of COMPASS, and the project could contribute significantly to the advancement of their careers by enabling them to gain prestigious postdoc and professor positions. COMPASS provided dedicated training to the staff of the partner institutions by the secondments itself, and via several invited seminars and one internal workshop. The project organized a symposium dedicated to the Colloidal Nanocrystal Materials at the Applied Nanotechnology and Nanoscience International Conference (ANNIC) in Berlin in 2018 that was open to the whole scientific community. The project strengthened the scientific community in Europe that work on nanomaterials for a sustainable future.
© COMPASS 2020
Colloidal Nanomaterials for Smart Applications
Call: H2020-MSCA-RISE-2015.
IIT's website uses the following types of cookies: browsing/session, analytics, functional and third party cookies. Users can choose whether or not to accept the use of cookies and access the website. By clicking on "Further Information", the full information notice on the types of cookies will be displayed and you will be able to choose whether or not to accept them whilst browsing on the website.
Further Information I Understand


text:
Property sales in Halton Village, Buckinghamshire
The average property price in Halton Village, Buckinghamshire is £951,000 which is £669,839 higher than the UK average of £281,161. Last year in 2021 property prices in Halton Village increased 0.9% from 2020.
19, May 2021 19, May 21 Dashwood, 15, Aylesbury, Buckinghamshire, HP22 5NS Detached £1,485,000 View sale history
4, February 2021 4, Feb 21 7, The Leys, Aylesbury, Buckinghamshire, HP22 5GH Terraced £417,000 View sale history
23, December 2020 23, Dec 20 Canal Cottage, , Aylesbury, Buckinghamshire, HP22 5NS Detached £1,135,000 View sale history
21, December 2020 21, Dec 20 4, The Leys, Aylesbury, Buckinghamshire, HP22 5GH Terraced £750,000 View sale history
22, January 2016 22, Jan 16 12, The Leys, Aylesbury, Buckinghamshire, HP22 5GH Terraced £190,000 View sale history
24, July 2014 24, Jul 14 1, The Leys, Aylesbury, Buckinghamshire, HP22 5GH Terraced £248,000 View sale history
17, July 2014 17, Jul 14 5, The Leys, Aylesbury, Buckinghamshire, HP22 5GH Semi-Detached £470,000 View sale history
11, April 2014 11, Apr 14 24, , Aylesbury, Buckinghamshire, HP22 5NS Semi-Detached £610,000 View sale history
7, September 2012 7, Sep 12 23, , Aylesbury, Buckinghamshire, HP22 5NS Semi-Detached £382,500 View sale history
8, December 2011 8, Dec 11 30b, Halton Lane, Aylesbury, Buckinghamshire, HP22 5NU Semi-Detached £407,500 View sale history
In Halton Village the most common property type purchased is Terraced with 50.0% of the total sales.
In Halton Village the most common build type purchased is Existing build with 100.0% of the total sales.
In Halton Village the most common estate type purchased is Freehold with 100.0% of the total sales.
Based on sales made in 2022, in the least expensive location to purchase a property in Halton Village with an average price of £.
Based on sales made in 2022, in the most expensive location to purchase a property in Halton Village with an average price of £
🚨 Crime statistics for Halton Village
📏 Halton Village surrounding area prices
View prices in Aylesbury
View prices in High Wycombe
View prices in Chesham
View prices in Amersham


text:
Q: Python: How to add RSA padding? I've been looking at most python crypto libraries, I've decided to use either PyCrypto or M2Crypto.
I've discarded ezPyCrypto because it only supports MD5 for signing and Keyczar because it's not mature enough.
So I've read that RSA is vulnerable to several attacks if the to-be-encrypted text (or signature hash) is not properly padded.
What does it mean?
Neither PyCrypto or M2Crypto mention anything about this and google didn't find anything relevant. Do these libraries automatically add the paddign? How can one tell?
If the answer to the above is no, what is considered proper padding?

A: PyCrypto doesn't add the mentioned padding.
M2Crypto instead does.
M2Crypto is built on top of openSSL, supports mostlyl everything you need, is still maintained and up to date while PyCrypto issues several deprecation warnings.

A: One of the reason for random padding might be that "from the book" RSA with low exponent (let's say 3) can be cracked really simply if the exact same message is sent to several people (three).
You'd therefore better make sure that you don't send the exact same message by applying some kind of random (yet inversible) transformation to your message before.
Maybe that's what thing padding is about !?
EDIT:
I looked on wikipedia. what I was talking about is called Hastad's attack.

A: I recently fought through figuring out encryption...this article helped alot in explaining what was ment by padding:
http://www.di-mgt.com.au/cryptopad.html
(method one seemed the easiest for me to implement)
I can share some code snippets if needed.
P.S. This file came in handy too in helping create secure keys (google for it) PBKDF2.py - PKCS#5 v2.0 Password-Based Key Derivation

A: Not entirely sure, but if you add a random component to your RSA message it prevents dictionary attacks

A: Firstly, you should be using AES, since it is the de-facto standard.
AES encrypts bytes in block-sizes of 16 bytes. Obviously this works fine for any large piece of data. But the last bit of it, obviously maybe lesser than 16 bytes.
For the last block, you'll need to pad it,and typical padding is done via PCKS7, which is pretty straight-forward.
Lets say you have a string: "icecream" as the last block.
"icecream" is 8 bytes, so you need another 8 bytes to make a block
So what you do is simply append the character 8(not '8') 8 times
"icecream\x08\x08\x08\x08\x08\x08\x08\x08"

Would be your resultant string. Now you go ahead an encrypt the data.
Remember that while decrypting, you'll need to catch this last block, and strip the padding before using it.

redpajama_set_name: RedPajamaStackExchange


Though Tennessee may be ranked fairly low on the list of gay-friendly states, the Memphis area has always had a reputation for being open-minded and accepting according to gay realtors Memphis. For this reason, many young gay and lesbian couples have a reason to seek to relocate to this area.
Memphis offers a very lively and active gay bar scene, including such famous places as The Pumping Station, The Backstreet Night Club, Crossroads, Dru's Place, The Jungle, and Madison Flame. Each bar caters primarily to a specific type of gay man, so the options for someone seeking to find a special someone are very diverse and varied.
For travelers seeking a place to settle down, the best places to do so in the Memphis, Tennessee area would be Germantown, Cooper-Young, Cordova, and Midtown.
The Cordova Area, near Memphis, was originally a very small farming community which only featured around 150 individuals. Over time, it would face a giant population boon to become one of the fastest growing neighborhoods in the entire state. The Cordova area, though open-minded, has a strong religious background and would probably be an area best-suited for older gay couples looking to settle down and lead a quiet family life.
Gay realtors Memphis report Germantown, Tennessee, not far from Memphis, is the ideal destination for the young LGBT community. Germantown's nightlife not only bustles with life and color from the city's many gay friendly bars and dance clubs, it also is chiefly dominated by retail and commercial outlets, which means that Germantown is a supreme shopping destination. This area may be suitable for the younger generation gay person who still enjoys parties, dancing, and has enough money to spend on dining out, theatre and luxurious, expensive clothing. Not only does Germantown offer a very relaxed lifestyle in terms of fun and entertainment, it also offers an alternative job market, since it is a fairly urbanized, non-industrial area.
Lastly, the Cooper-Young neighborhood of Memphis Tennessee could be considered the premiere art district in the area. Gay realtors Memphis tell us every fall, the neighborhood kicks off seasonal celebrations with a large-scale event it calls the Cooper-Young Festival. This festival not only features live music and dancing, but it also is host to a large number of arts vendors, crafts vendors, and features a staggering 4-mile marathon. It is one of the Memphis area's largest festivals, and would be a great place for the younger, art-minded LGBT member to find a suitable place to call home.
Memphis is a large musical center, home of Sun Studio where Elvis recorded his first song ever.


text:
Treatment of soft drink bottling wastewater from bench-scale treatability to full-scale operation
Section 9. FOOD WASTES
TREATMENT OF SOFT DRINK BOTTLING WASTEWATER
FROM BENCH-SCALE TREATABILITY TO FULL-SCALE OPERATION
Frederic C. Blanc, Professor
James C. O'Shaughnessy, Assistant Professor
Steven H. Corr, Vice President
Peter M. Smith, Engineer
Weston & Sampson Engineers, Inc.
Wakefield, Massachusetts 01880
When a soft drink bottling plant on Cape Cod decided to construct a new bottling plant in Sandwich, Massachusetts, a wastewater treatment and disposal system was required by the Commonwealth
of Massachusetts since there was no municipal sewer system available. This paper presents the treatability investigation, design, and construction of the wastewater facility for this soft drink bottling
plant.
The bottling plant bottles Cola, Diet Cola, Orange Soda, Root Beer, and other flavors for distribution on Cape Cod and surrounding areas. The Company bottles 2 liter, 32 ounce, 16 ounce and 10
ounce returnable bottles, and 5-gallon transfer tanks. All of the operations are bottling operations.
No canning is done. The process is operated year-round, normally one 8-hour shift per day. However,
in the summer when production more than doubles, this may be increased to 10-hours per day.
There are two production bottling lines located in the plant: Line Number 1 is for all sizes except 2
liter while Line Number 2 is for the 2 liter size only. Transfer tanks are washed and filled in a separate
area. All of the existing equipment was moved to the new facility. The wastewater sampling and characterization was conducted at the old facility.
The Bottling Process and Wastewater Generation
A process schematic, with the bottling operation, uses of water, and the sources of wastewater, is
shown on Figure 1. Wastewater is generated in several locations in the operation.
Production Line Number 1 fills new and returnable bottles. When new bottles are being filled, a
prerinse with a flow rate of approximately 10 gallons per minute (gpm) precedes the filling operation.
Additional wastewater is generated by soap, conveyor belt lubricant and the disposal of the contents
of improperly filled bottles.
When returnable bottles are on Production Line 1, the bottles pass through the bottle washing
machine. The machine washes the returnable bottles in a hot caustic solution which includes the following: 1) Pre-rinse; 2) Caustic active soak; 3) Rinse; 4) Warm water rinse; and 5) Final fresh water
rinse. The rinses from the bottle washer are discharged to the floor drain at a flow rate of approximately 35 gpm.
Transfer containers are 5-gallon containers filled with either premixed soft drink or a soft drink
concentrate to be mixed with carbonated water in a soft drink dispensing machine. Containers are
reused and washed in a special washing machine prior to refilling. Rinse water from this machine is
discharged as wastewater. In addition, any contents remaining in a returned can are emptied to the
floor drain.
Production Line Number 2 bottles only new 2-liter bottles in the same manner as Production Line
Number 1. Bottles are pre-rinsed and post-rinsed after bottling for a total flow of 15 to 20 gpm.
Title Treatment of soft drink bottling wastewater from bench-scale treatability to full-scale operation
Author Blanc, Frederic C.
O'Shaughnessy, James C.
Corr, Steven H.
Smith, Peter M.
Conference Title Proceedings of the 39th Industrial Waste Conference
Conference Front Matter (copy and paste) http://e-archives.lib.purdue.edu/u?/engext,35769
Rights Statement Digital copyright Purdue University. All rights reserved.
Transcript Section 9. FOOD WASTES TREATMENT OF SOFT DRINK BOTTLING WASTEWATER FROM BENCH-SCALE TREATABILITY TO FULL-SCALE OPERATION Frederic C. Blanc, Professor James C. O'Shaughnessy, Assistant Professor Department of Civil Engineering Northeastern University Boston, Massachusetts 02115 Steven H. Corr, Vice President Peter M. Smith, Engineer Weston & Sampson Engineers, Inc. Wakefield, Massachusetts 01880 INTRODUCTION When a soft drink bottling plant on Cape Cod decided to construct a new bottling plant in Sandwich, Massachusetts, a wastewater treatment and disposal system was required by the Commonwealth of Massachusetts since there was no municipal sewer system available. This paper presents the treatability investigation, design, and construction of the wastewater facility for this soft drink bottling plant. The bottling plant bottles Cola, Diet Cola, Orange Soda, Root Beer, and other flavors for distribution on Cape Cod and surrounding areas. The Company bottles 2 liter, 32 ounce, 16 ounce and 10 ounce returnable bottles, and 5-gallon transfer tanks. All of the operations are bottling operations. No canning is done. The process is operated year-round, normally one 8-hour shift per day. However, in the summer when production more than doubles, this may be increased to 10-hours per day. There are two production bottling lines located in the plant: Line Number 1 is for all sizes except 2 liter while Line Number 2 is for the 2 liter size only. Transfer tanks are washed and filled in a separate area. All of the existing equipment was moved to the new facility. The wastewater sampling and characterization was conducted at the old facility. The Bottling Process and Wastewater Generation A process schematic, with the bottling operation, uses of water, and the sources of wastewater, is shown on Figure 1. Wastewater is generated in several locations in the operation. Production Line Number 1 fills new and returnable bottles. When new bottles are being filled, a prerinse with a flow rate of approximately 10 gallons per minute (gpm) precedes the filling operation. Additional wastewater is generated by soap, conveyor belt lubricant and the disposal of the contents of improperly filled bottles. When returnable bottles are on Production Line 1, the bottles pass through the bottle washing machine. The machine washes the returnable bottles in a hot caustic solution which includes the following: 1) Pre-rinse; 2) Caustic active soak; 3) Rinse; 4) Warm water rinse; and 5) Final fresh water rinse. The rinses from the bottle washer are discharged to the floor drain at a flow rate of approximately 35 gpm. Transfer containers are 5-gallon containers filled with either premixed soft drink or a soft drink concentrate to be mixed with carbonated water in a soft drink dispensing machine. Containers are reused and washed in a special washing machine prior to refilling. Rinse water from this machine is discharged as wastewater. In addition, any contents remaining in a returned can are emptied to the floor drain. Production Line Number 2 bottles only new 2-liter bottles in the same manner as Production Line Number 1. Bottles are pre-rinsed and post-rinsed after bottling for a total flow of 15 to 20 gpm. 243
Treatment of soft drink bottling wastewater from...


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