Art . War . "We Protest: Iowa Speaks Out"
In Spring 2006, the Hearst Center for the Arts, in Cedar Falls, Iowa, announced the exhibition "First Fifty 2006: Works by Cedar Valley Artists". The first fifty artists to arrive at the center with their work by 8 a.m. on May 11, 2006, would be included in the show.
Some of us saw this as an opportunity to speak out about the war in Iraq. We asked a number of area artists to consider creating such a work and to arrive early on the 11th to be among the first fifty in line. The show opening on May 20, 2006, was very well attended.
Since this event we have felt compelled to continue our protest by offering these works for exhibition in the hopes of having our voices heard. To date the work has been shown at Altered Esthetics in Minneapolis; the Witter in Storm Lake, Iowa; Luther College in Decorah, Iowa; and United World College, in Montezuma, New Mexico.


text:
\section{Introduction}
The hydrodynamic model was first derived by Bl{\o}tekj{\ae}r \cite{Bl70} for electrons in a semiconductor. After appropriate simplification the one-dimensional time-dependent system in the  isentropic case reads:
\begin{equation}\label{1.1}
\left\{
\begin{split}
&n_t+(nu)_x=0,\\
&(nu)_t+\left(nu^2+p(n)\right)_x=nE-\frac{nu}{\tau },\\
&E_x=n-b(x),
\end{split}
\right.
\end{equation}
where $n(x,t)$, $u(x,t)$ and $E(x,t)$ denote the electron density, velocity, and electric field respectively. The given function $p=p(n)$ is the pressure-density relation on which a commonly used hypothesis is
\begin{equation*}
p(n)=Tn^\gamma,
\end{equation*}
where $T>0$ is Boltzmann's constant and $\gamma\ge1$ is the adiabatic exponent. The constant parameter $\tau>0$ is the momentum relaxation time. The given background density $b(x)>0$ is called the doping profile. The hydrodynamic model \eqref{1.1} is also called Euler-Poisson equations with semiconductor effect. For more details we refer  to treatises \cite{MRS90,ZH16} and references therein.

In this paper, the main focus is on the isothermal steady-state flows satisfying equations
\begin{equation}\label{hu17}
\left\{
\begin{split}
&J\equiv\text{constant},\\
&\left(\frac{J^2}{n}+p(n)\right)_x=nE-\frac{J}{\tau},\\
&E_x=n-b(x),
\end{split}
\right.
\end{equation}
where $J=nu$ stands for the current density, and $p(n)=Tn$ corresponds to the isothermal ansatz. By the terminology from gas dynamics, we call $c:=\sqrt{P'(n)}=\sqrt T>0$ the speed of sound. The flow is referred to as subsonic, sonic or supersonic provided the velocity satisfies
\begin{equation}\label{1.5}
u<c,\quad u=c\quad\text{or}\quad u>c,\quad\text{respectively}.
\end{equation}
For convenience of notation, we introduce
\begin{equation}\label{1.2*}
\alpha=\frac{1}{\tau},~~\mbox{the reciprocal of the momentum relaxation time}.
\end{equation}
Without loss of generality, we set
\begin{equation}\label{z}
T=1 \mbox{ and } J=1,
\end{equation}
 thus the system \eqref{hu17} is equivalently reduced to the system
\begin{equation}\label{1.6}
\left\{
\begin{split}
&\left(1-\frac{1}{n^2}\right)n_x=nE-\alpha,\\
&E_x=n-b(x).
\end{split}
\right.
\end{equation}
From \eqref{1.5} and \eqref{z}, it is easy to see that the flow is subsonic if $n>1$, sonic if $n=1$, or supersonic if $0<n<1$. By virtue of \eqref{1.2*}, we call the system \eqref{1.6} the Euler-Poisson equations with the semiconductor effect if $\alpha>0$, and without the semiconductor effect if $\alpha=0$, respectively. Throughout this paper, we are interested in the system \eqref{1.6} in the open interval $(0,1)$, which is subjected to the sonic boundary condition
\begin{equation}\label{1.7}
n(0)=n(1)=1.
\end{equation}
We also assume that the doping profile $b(x)$ is of class $C[0,1]$, satisfying the subsonic condition $b(x)>1$ on $[0,1]$. For simplicity of notation, its infimum and supremum over $[0,1]$ is denoted by
\[
\underline{b}:=\inf_{x\in[0,1]}b(x) \mbox{ and } \bar{b}:=\sup_{x\in[0,1]}b(x),
\]
 respectively.

Over the past three decades, major advances in the mathematical theory of steady-state Euler-Poisson equations with/without the semiconductor effect have been made by many authors. In what follows, we just list several results which are closely linked to the present paper.

For the purely subsonic steady-state flows,  in 1990, Degond
\textit{et al.}\cite{DM90} first proved the existence of the subsonic solution
to the one-dimensional steady-state Euler-Poisson with the
semiconductor effect when its boundary states belongs to the
subsonic region. Subsequently, Degond \textit{et al.}\cite{DM93} further
showed the existence and local uniqueness of irrotational subsonic
flows to the three-dimensional steady-state semiconductor
hydrodynamic model under a smallness assumptions on the data. Along
this line of research, the steady-state subsonic flows with and
without the semiconductor effect were investigated in various
physical boundary conditions and different dimensions
\cite{FI97,GS05,NS07,BDX14}. As for the purely supersonic
steady-state flows, Peng \textit{et al.}\cite{PV06} established the existence
and uniqueness of the supersonic solutions with the semiconductor
effect, which correspond to a large current density.

Note that the system \eqref{hu17} or \eqref{1.6} will be degenerate
at the sonic state, thus the study on the transonic solutions and
various steady states satisfying the sonic boundary condition
becomes very difficult. Ascher \textit{et al.}\cite{AMPS91} first examined the
existence of the transonic solution to the one-dimensional
isentropic Euler-Poisson equations without and with the
semiconductor effect when the doping profile is a supersonic
constant, and then Rosini \cite{Ro05} extended this work to the
non-isentropic case by the analysis of phase plane. When the doping
profile is non-constant, Gamba \cite{Ga92,GM96} investigated the
one-dimensional and two-dimensional transonic solutions with shocks,
respectively. However, these transonic solutions yield boundary
layers because they are constructed as the limits of vanishing
viscosity. Luo \textit{et al.}\cite{LX12,LRXX11} further considered the
one-dimensional Euler-Poisson equations without the semiconductor
effect, under the restriction that boundary data are far from the
sonic state and the doping profile is either a subsonic constant or
a supersonic constant, a comprehensive analysis on the structure and
classification of steady states was carried out in \cite{LX12} by
using the analysis of phase plane. Meanwhile, both structural and
dynamical stability of steady transonic shock solutions was obtained
in \cite{LRXX11}.

What if the sonic state appears in the solutions? As we have seen,
all the existing works introduced above cannot answer this question.
Even the works regarding transonic shocks cannot radically answer it
either because the two different phase states are connected by the
jump of shocks satisfying the  Rankine-Hugoniot condition and
entropy condition, avoiding the degeneracy caused by the sonic
state. Recently, Li \textit{et al.}\cite{LMZZ17,LMZZ18} systematically
explored the critical case, that is, the one-dimensional
semiconductor Euler-Poisson equations with the sonic boundary
condition. The existence, nonexistence and classification of all
types of physical steady states to this critical boundary-value
problem was obtained for the subsonic doping profile in
\cite{LMZZ17} and supersonic doping profile in \cite{LMZZ18}. More
precisely, in \cite{LMZZ17}, the authors proved that the critical
boundary-value problem admits a unique subsonic solution, at least
one supersonic solution, infinitely many transonic shocks if
$\alpha\ll1$, and infinitely many transonic $C^1$-smooth solutions
if $\alpha\gg1$; in \cite{LMZZ18}, the authors showed the
nonexistence of all types of physical steady states to the critical
boundary-value problem assuming that the doping profile is small
enough and $\alpha\gg1$, and they also discussed the existence of
supersonic and transonic shock solutions under the hypothesis that
the doping profile is close to the sonic state and $\alpha\ll1$.
Inspired by the groundbreaking works \cite{LMZZ17,LMZZ18}, there is
a series of interesting generalizations into the transonic doping
profile case in \cite{CMZZ20}, the case of transonic
$C^\infty$-smooth steady states in \cite{WMZZ21}, the
multi-dimensional cases in \cite{CMZZ21,CMZZ22}, and even the
bipolar case \cite{MMZ20}.

Compared with the existence theory in the critical case,
there are rather few works on the study of stability (both structural
 and dynamical stability) in the critical case due to the interior
  or boundary degeneracy. To the best of our knowledge,
  Feng \textit{et al.}\cite{FMZ22} first demonstrated the structural
  stability of the $C^1$-smooth transonic steady states with
  respect to the small perturbation of both the supersonic
  constant doping profiles and non-degenerate boundary data.
  In \cite{FMZ22}, for supersonic doping profile, the authors
   also discussed the structural stability and linear dynamic
   instability of the transonic steady states with the non-degenerate
   boundary data under some appropriate hypotheses. Thus far, the most
   difficult part of the critical case, namely, the problem about the
    structural and dynamical stability of all types of physical solutions
     with the sonic boundary condition for the non-constant subsonic
     doping profile (for the existence theory, see \cite{LMZZ17}),
     is still open. To thoroughly solve this problem is full of challenges,
     owing to the boundary degeneracy. However, in the present paper,
     we intend to shed new light on this problem.

The purpose of this paper is to show that interior subsonic
solutions to the system \eqref{1.6} with the sonic boundary
condition \eqref{1.7} are structurally stable if we propose
a monotonicity restriction on subsonic doping profiles (see Theorem \ref{T2}).

This paper is organized as follows. Some necessary preliminaries
and the main result are stated in Section \ref{s2}. The proof
of the main result, Theorem \ref{T2}, is given in Section \ref{s3}.

\section{Preliminaries and the main result}\label{s2}
In this section we shall present the main result.
Before proceeding, we first give the important preliminaries
from the foregoing research \cite{LMZZ17}. First of all,
we recall the definition of the interior subsonic solution.
\begin{defi}\label{def}
We say a pair of functions $(n,E)(x)$ is an interior subsonic
solution of the boundary value problem \eqref{1.6}\&\eqref{1.7}
provided \textup{(i)} $(n-1)^2\in H_0^1(0,1)$, \textup{(ii)} $n(x)>1$,
for all $x\in(0,1)$, \textup{(iii)} $n(0)=n(1)=1$, \textup{(iv)}
the following equality holds for all test functions $\varphi\in H_0^1(0,1)$,
\begin{equation}\label{1.9}
\int_0^1\left(\frac{1}{n}-\frac{1}{n^3}\right)n_x\varphi_xdx+\alpha\int_0^1\frac{\varphi _x}{n}dx+\int_0^1\left(n-b\right)\varphi dx=0,
\end{equation}
and \textup{(v)} $E(x)$ is given by
\begin{equation}\label{E}
E(x)=\alpha +\int_0^x\left(n(y)-b(y)\right)dy.
\end{equation}
\end{defi}
In addition, we continue to recall the existence and uniqueness
of interior subsonic solutions, which is excerpted from the first
part of Theorem 1.3 in \cite{LMZZ17}.

\begin{prop}[Existence theory in \cite{LMZZ17}]\label{T1}
Suppose that the doping profile $b\in L^\infty(0,1)$ is subsonic
such that $\underline{b}>1$. Then for any $\alpha\in[0,\infty)$
the boundary value problem \eqref{1.6}\&\eqref{1.7} admits a unique
interior subsonic solution $(n,E)\in C^{\frac{1}{2}}[0,1]\times H^1(0, 1)$
satisfying the boundedness
\begin{equation}\label{ulb}
1+m\sin(\pi x)\le n(x)\le\bar{b}, \quad x\in [0,1],
\end{equation}
and the boundary behavior at endpoints
\begin{equation}\label{E(1)}
E(0)=\alpha,\quad E(1)<\alpha,
\end{equation}
\begin{equation}\label{L14}
\left\{
\begin{split}
& C_1(1-x)^\frac{1}{2}\le n(x)-1\le C_2(1-x)^\frac{1}{2},\\
& -C_3(1-x)^{-\frac{1}{2}}\le n_x(x)\le-C_4(1-x)^{-\frac{1}{2}},
\end{split}
\right. \quad\text{for}\ x\ \text{near}\ 1,
\end{equation}
where $m=m(\alpha, \underline{b})>0$, $C_2>C_1>0$ and $C_3>C_4>0$ are
certain uniform estimate constants.
\end{prop}

\begin{rem}
Note that the degeneracy of the boundary value problem
\eqref{1.6}\&\eqref{1.7} occurs merely on the boundary.
Thus, if we assume that the doping profile has relatively
higher-order regularity, say $b\in C[0,1]$, then by virtue
of the standard theory for elliptic interior regularity and
Sobolev's embedding theorem, the corresponding interior
subsonic solution $(n,E)$ is actually of class
$\left(C^1(0,1)\cap C^{\frac{1}{2}}[0,1]\right) \times C^1[0,1]$.
This fact will be tacitly exploited hereafter.
\end{rem}

We are now in a position to formulate the main result in the present paper.
\begin{theo}[Structural stability of interior subsonic solutions]\label{T2}
Assume that $b_1, b_2\in C[0,1]$ are subsonic such that $b_1(x)\ge b_2(x)>1$
for all $x\in[0,1]$. For $i=1,2$, let $(n_i,E_i)(x)$ denote the
interior subsonic solution relative to the doping profile $b_i$,
respectively. Then the two interior subsonic solutions are
structurally stable to one another in the sense that
\begin{equation}\label{1.10**}
\|n_1-n_2\|_{C[0,1]}+\|(1-x)^{\frac{1}{2}}(n_1-n_2)_x\|_{C[0,1]}+\|E_1-E_2\|_{C^1[0,1]}\le C\|b_1-b_2\|_{C[0,1]},
\end{equation}
where $C>0$ is a certain constant independent of $\|b_1-b_2\|_{C[0,1]}$.
\end{theo}

We conclude this section with a brief sketch of the strategy that
underlies the proof of our main result. Due to the boundary
degeneracy of interior subsonic solutions, the study of their
globally structural stability over the entire interval $[0,1]$
becomes sophisticated and challenging. Therefore, we shall have
to divide the whole interval $[0,1]$ into three domains as follows:
\begin{equation*}
[0,1]=[0,\delta)\cup[\delta,1-\delta]\cup(1-\delta,\delta],
\end{equation*}
where the intrinsic segmentation constant $\delta>0$ would be
appropriately determined (see Lemma \ref{L5}), and we will also
have to establish structural stability estimates separately on
their respective domains in the following order: (i) near the
left endpoint $x=0$; (ii) near the right endpoint $x=1$; (iii)
on the middle domain. This strategy is feasible because we have
discovered the following facts:
\begin{enumerate}
\item the local singularity analysis reveals that the plausible
singularity at the left endpoint $x=0$ is removable (see Lemma \ref{L1});
based on this, we are able to establish the local structural stability
 estimate on an inherent neighborhood $[0,\delta_0)$ by the
 monotonicity argument. The main point is that both the radius
 $\delta_0$ and the positive estimate constant are independent
 of $\|b_1-b_2\|_{C[0,1]}$ (see Lemma \ref{L3}). This is the
 reason why this type of neighborhood is referred to as ``to be intrinsic
 or inherent''. This sort of tacit convention will be used throughout
 the present paper.

\item the local weighted singularity analysis discloses that the
genuine singularity at the right endpoint $x=1$ can be well controlled
by the $(1-x)^{\frac{1}{2}}$-weight (see Lemma \ref{L4-1}); thus,
the monotonicity argument further ensures that the local weighted
structural stability holds on an intrinsic neighborhood
$(1-\delta_1,1]$ (see Lemma \ref{L4}).

\item the remaining part constitutes the middle domain, which is
regular as to the structural stability (see Lemma \ref{L5}).
\end{enumerate}

It is worth mentioning that the monotonicity argument has been playing
a crucial role in establishing structural stability estimates near
both endpoints. The principle behind the monotonicity argument
is given by Lemma \ref{Lcp}. From a technical point of view,
the monotonicity argument is useful, but only at the cost of
adding an extra restriction $b_1(x)\ge b_2(x)$ on $[0,1]$.
How to get rid of this restriction is a tough question, and
we will explore it in the future study.

\section{Proof of Theorem \ref{T2}}\label{s3}
This section is devoted to proving our main result. In order to make
the line of reasoning accessible to the reader,
the proof will be
divided into a sequence of lemmas.

We let $(n_i,E_i)(x)$ denote the interior subsonic solution
corresponding to the subsonic doping profile $b_i(x)>1$, satisfying
the sonic boundary value problem
\begin{equation}\label{1}
\left\{
\begin{split}
&\left(1-\frac{1}{n_i^2}\right)n_{ix}=n_iE_i-\alpha,\\
&E_{ix}=n_i-b_i(x), \quad x\in (0,1), \\
&n_i(0)=n_i(1)=1,
\end{split}
\right. \quad\text{for}\ i=1,2, \ \text{respectively}.
\end{equation}

First of all, we adapt the comparison principle in
\cite{LMZZ17}(Lemma 2.2, P4773) for use with two doping profiles
and their corresponding interior subsonic solutions, which is the
basis of the monotonicity argument in studying the structural
stability near endpoints.

\vspace{1.5mm}
\begin{lem}[Comparison principle]\label{Lcp}
Let the doping profiles $b_1, b_2\in C[0,1]$.
If $b_1(x)\ge b_2(x)>1$ on $[0,1]$. Then
\begin{equation}\label{1.10*}
n_1(x)\ge n_2(x),\ \text{on}\ [0,1].
\end{equation}
\end{lem}

\vspace{3mm}
\noindent\textbf{Proof.}
According to the relevant arguments from \cite{LMZZ17}(Equation (17), P4773),
for $i=1,2$, since $(n_i,E_i)$ is the interior subsonic solution,
thereby having the approximate solution sequence
$\{n_{ij}\}_{0<j<1}\subset C^1[0,1]$ satisfying the weak form
\begin{equation}\label{wf}
\int_0^1 A(n_{ij}, n_{ijx})\varphi_xdx+\int_0^1 (n_{ij}-b_i)\varphi dx=0,\quad\forall\varphi\in H_0^1(0,1),
\end{equation}
where
\begin{equation*}\label{1.12}
A(z,p):=\left(\frac{1}{z}-\frac{j^2}{z^3}\right)p+\alpha\frac{j}{z}.
\end{equation*}

Subtracting $\eqref{wf}|_{i=1}$ from $\eqref{wf}|_{i=2}$,
for all nonnegative test functions $\varphi\in H_0^1(0,1)$, we have
\begin{equation}\label{1.13}
\int_0^1 \left(A(n_{2j},n_{2jx})-A(n_{1j},n_{1jx})\right)\varphi_xdx+\int_0^1 \left(n_{2j}-n_{1j}\right)\varphi dx=\int_0^1 \left(b_2-b_1\right)\varphi dx\le0,
\end{equation}
where we have used the assumption that $b_1(x)\ge b_2(x)$ on $[0,1]$
in the last inequality. This is exactly the crucial Equation (19) in
\cite{LMZZ17}, the same result therefore applies to \eqref{1.13}
provided we simply imitate the remaining arguments in Lemma 2.2
of \cite{LMZZ17}. That is,
\begin{equation}\label{1.14}
n_{1j}(x)\ge n_{2j}(x),\ \text{on}\ [0,1], \ \text{for}\ 0<j<1.
\end{equation}

Now the monotonicity relation \eqref{Lcp} follows after a passage to
the limit as $j\to1^-$ on both sides of the inequality \eqref{1.14}.\qed
\vspace{3mm}

In addition, for $i=1,2$, we set about analyzing the boundary behavior of
the first-order derivative of $n_{i}(x)$ at the left endpoint $x=0$.
 It seems plausible that the singularity should have appeared there,
 as a matter of fact this ``fake'' singularity at $x=0$ is removable
 because of $E_i(0)=\alpha$.

\vspace{1.5mm}
\begin{lem}\label{L1}
Suppose that $b_i, i=1,2$ satisfy the same conditions in Lemma \ref{Lcp},
and $\alpha\ge2\sqrt{2}\max\{\sqrt{b_1(0)-1}, \sqrt{b_2(0)-1}\}$. Then
\begin{equation}\label{3}
\lim_{x\to0^+}n_{ix}(x)=\frac{1}{4}\left(\alpha-\sqrt{\alpha^2-8\left(b_i(0)-1\right)}\right)=:A_i>0,\quad i=1,2.
\end{equation}
\end{lem}

\vspace{3mm}
\noindent\textbf{Proof.}
In much the same way as in \cite{LMZZ17}(Theorem 5.6, P4802),
owing to $n_i(0)=1$ and $E_i(0)=\alpha$, it is easy to see
that $\lim_{x\to0^+}n_{ix}(x)$ exists by the monotone convergence
argument. Then from the first equation of \eqref{1}, we have
\begin{equation*}
n _{ix}=\frac{ E_i n _i^2 }{ n _i  + 1 } + \frac{\left(E_i-\alpha\right) n _i^2 }{ \left(  n _i  - 1  \right)\left( n _i  + 1  \right)}, \quad\text{in}\ (0,1).
\end{equation*}

Noting that $n_i(0)=1$ and $E_i(0)=\alpha$, it follows from the
L'Hospital Rule that
\begin{align*}
A_i &=\lim_{x\to0^+}n_{ix}(x)=\lim_{x\to0^+}\frac{E_in_i^2}{n_i+1}+\lim_{x\to0^+}\frac{\left(E_i-\alpha\right)n_i^2}{\left(n_i-1\right)\left(n_i+1\right)} \\
&=\frac{\alpha}{2}+\frac{1}{2}\lim_{x\to0^+}\frac{\left(E_i-\alpha\right)_x}{\left(n_i-1\right)_x}=\frac{\alpha}{2}+\frac{1}{2}\lim_{x\to0^+}\frac{E_{ix}}{n_{ix}}\\
&=\frac{\alpha}{2}+\frac{1}{2}\lim_{x\to0^+}\frac{n_i(x)-b_i(x)}{n_{ix}}\\
&=\frac{\alpha}{2}+\frac{1-b_i(0)}{2A_i},
\end{align*}
which in turn implies that
\begin{equation*}
A_i=\frac{1}{4}\left(\alpha-\sqrt{\alpha^2-8\left(b_i(0)-1\right)}\right)=\frac{2\left(b_i(0)-1\right)}{\alpha+\sqrt{\alpha^2-8\left(b_i(0)-1\right)}}=O(\frac{1}{\alpha}),
\end{equation*}
or
\begin{equation*}
A_i=\frac{1}{4}\left(\alpha+\sqrt{\alpha^2-8\left(b_i(0)-1\right)}\right)=O(\alpha).
\end{equation*}

According to the local singularity analysis in \cite{LMZZ17}(Lemma 5.3, P4796),
we know that in a small neighborhood of $x=0$, the drastic change of the
density component $n_i(x)$ of the interior subsonic solution is impossible
when $\alpha$ is suitably large. Therefore, we have to choose the former
root as the limit value of $\lim_{x\to0^+}n_{ix}(x)$, and the latter one is
the extraneous root.\qed
\vspace{3mm}

Based on Proposition \ref{T1} and Lemmas \ref{Lcp}$\sim$\ref{L1}, we are
now preparing to establish the local structural stability of interior
subsonic solutions to the boundary value problem \eqref{1.6}\&\eqref{1.7}
 on an intrinsic neighborhood of the left endpoint $x=0$.

\vspace{1.5mm}
\begin{lem}[Local structural stability estimate near $x=0$]\label{L3}
Under the same conditions in Lemma \ref{L1}. There exist two positive
constants $\delta_0\in(0,\frac{1}{2})$ and $C>0$ independent of
 $\|b_1-b_2\|_{C[0, 1]}$ such that
\begin{equation}\label{key-1}
\|n_1-n_2\|_{C^1[0,\delta_0)}+\|E_1-E_2\|_{C^1[0,\delta_0)}\le C\| b_1-b_2\|_{C[0,1]}.
\end{equation}
\end{lem}

\vspace{3mm}
\noindent\textbf{Proof.}
Firstly, in light of Lemma \ref{Lcp}, it is clear that the following
monotonicity relation holds,
\begin{equation}\label{mr}
\frac{n_1^3}{n_1+1}\ge\frac{n_2^3}{n_2+1},\quad\forall x\in[0,1].
\end{equation}

Next, for simplicity, we set $\tilde{E_i}:=E_i-\frac{\alpha}{n_i}$.
Multiplying Equation $\eqref{1}_1$ by $\frac{n_i^2}{n_i^2-1}$, we have
\begin{equation}\label{h1}
n_{ix}=\frac{\tilde{E}_i n_i^3}{n_i^2-1}, \quad  i=1,2.
\end{equation}
Taking the difference of Equations $\eqref{h1}|_{i=1}$ and $\eqref{h1}|_{i=2}$,
near $x=0$, we compute together with the monotonicity relation \eqref{mr} that
\begin{align}\label{33}
(n_1-n_2)_x=&\frac{\tilde{E}_1 n_1^3}{n_1^2-1}-\frac{\tilde{E}_2 n_2^3}{n_2^2-1}\\
=&\frac{n_1^3}{n_1+1}\frac{\tilde{E}_1}{n_1-1}-\frac{n_2^3}{n_2+1}\frac{\tilde{E}_1}{n_1-1}+\frac{n_2^3}{n_2+1}\frac{\tilde{E}_1}{n_1-1}-\frac{n_2^3}{n_2+1}\frac{\tilde{E}_2}{n_2 - 1}\notag\\
=&\frac{\tilde{E}_1}{n_1-1}\left(\frac{n_1^3}{n_1+1}-\frac{n_2^3}{n_2+1}\right)+\frac{n_2^3}{n_2+1}\left(\frac{\tilde{E}_1}{n_1-1}-\frac{\tilde{E}_2}{n_2-1}\right)\notag\\
\leq&M_0\alpha\left(\frac{n_1^3}{n_1+1}-\frac{n_2^3}{n_2+1}\right)+\frac{n_2^3}{n_2+1}M_0\|b_1-b_2\|_{C[0, 1]},\notag\\
\leq&C(n_1-n_2)+C\|b_1-b_2\|_{C[0, 1]},\quad x\in[0,\delta_0), \notag
\end{align}
where we have used the fact that there exist two positive constants
$\delta_0\in(0,\frac{1}{2})$ and $M_0>0$ independent of $\|b_1-b_2\|_{C[0, 1]}$ such that
\begin{equation}\label{0227}
\frac{\tilde{E}_1}{n_1- 1}(x)\leq M_0\alpha,\quad\text{and}\quad \left( \frac{\tilde{E}_1}{n_1- 1}-\frac{\tilde{E}_2}{n_2- 1}\right) (x)\leq M_0\|b_1-b_2\|_{C[0, 1]},\quad x\in[0,\delta_0).
\end{equation}

To prove that the crucial estimate \eqref{0227} on a certain intrinsic
neighborhood $[0,\delta_0)$ holds, we assume for the sake of contradiction
that for any $\delta\in(0,\frac{1}{2})$ and $M>0$, there exists
$x_\delta\in[0,\delta)$ such that
\begin{equation}\label{C0227}
\frac{\tilde{E}_1}{n_1-1}(x_\delta)>M\alpha, \quad\text{or}\quad \left( \frac{\tilde{E}_1}{n_1-1}-\frac{\tilde{E}_2}{n_2-1}\right) (x_\delta)> M\|b_1-b_2\|_{C[0, 1]}.
\end{equation}
Particularly, we take $\delta=\frac{1}{k}, k=3,4,5,\cdots$, for any $M>0$,
there is $x_k\in[0,\frac{1}{k})$ such that
\begin{equation*}
\frac{\tilde{E}_1}{n_1-1}(x_k)>M\alpha, \quad\text{or}\quad \left( \frac{\tilde{E}_1}{n_1-1}-\frac{\tilde{E}_2}{n_2-1}\right) (x_k)> M\|b_1-b_2\|_{C[0, 1]},
\end{equation*}
which implies that
\begin{equation}\label{0227-1}
\varliminf_{x_k\to0^+}\frac{\tilde{E}_1}{n_1-1}(x_k)\geq M\alpha,
\end{equation}
or
\begin{equation}\label{0227-2}
\varliminf_{x_k\to0^+}\left(\frac{\tilde{E}_1}{n_1-1}-\frac{\tilde{E}_2}{n_2-1}\right)(x_k)\geq M\|b_1-b_2\|_{C[0, 1]}.
\end{equation}
Combining the boundary behavior \eqref{E(1)}, the L'Hospital Rule,
Equation $\eqref{1}_2$ and Lemma \ref{L1}, we calculate
\begin{align}\label{0227-3}
\lim_{x\to0^+}\frac{\tilde{E}_i}{n_i-1}(x)=&\lim_{x\to0^+}\frac{E_i-\frac{\alpha}{n_i}}{n_i-1}(x)\\
=&\lim_{x\to0^+}\frac{E_i(x)-E_i(0)+\alpha-\frac{\alpha}{n_i(x)}}{n_i(x)-1} \notag\\
=&\lim_{x\to0^+}\frac{E_i(x)-E_i(0)}{n_i(x)-1}+\lim_{x\to0^+}\frac{\alpha}{n_i(x)}\notag\\
=&\lim_{x\to0^+}\frac{n_i-b_i}{n_{ix}}+\alpha \notag\\
=&\frac{1-b_i(0)}{A_i}+\alpha<\alpha, \quad i=1,2,\notag
\end{align}
and
\begin{align}\label{0227-4}
\lim_{x\to0^+}\left( \frac{\tilde{E}_1}{n_1-1}-\frac{\tilde{E}_2}{n_2-1}\right)(x)=&\frac{b_2(0)-1}{A_2}-\frac{b_1(0)-1}{A_1}\\
=&\frac{1}{2}\left(\sqrt{\alpha^2-8(b_2(0)-1)}-\sqrt{\alpha ^2-8(b_1(0)-1)}\right) \notag\\
=&\frac{2}{\sqrt{\alpha^2-8(\eta-1)}}\left(b_1(0)-b_2(0)\right) \notag\\
\leq&\tilde{C}_0\|b_1-b_2\|_{C[0, 1]},\notag
\end{align}
where $\eta\in(b_2(0), b_1(0))$. Furthermore,  we note that the constant $M$
in \eqref{0227-1} and \eqref{0227-2} can be chosen arbitrarily. Consequently,
if we take $M=2$ in \eqref{0227-1}, together with \eqref{0227-3}, we obtain
the contradiction that $2\alpha<\alpha$; if we take $M=2\tilde{C}_0$ in
\eqref{0227-2}, combined with \eqref{0227-4}, we have the contradiction $2\leq1$.

Based on the local estimate \eqref{33}, we continue establishing the structural
stability locally on the intrinsic neighborhood $[0,\delta_0)$. To this end,
we multiply through the inequality \eqref{33} by $n_1-n_2$ and calculate
\begin{equation}\label{26}
\frac{d}{dx}(  n_1 -n_2)^2(x)\le C( n_1-n_2)^2(x)+C\|b_1-b_2\|_{C[0,1]}^2,\quad x\in[0, \delta_0),
\end{equation}
where we have used Lemma \ref{Lcp} and Cauchy's inequality. By Gronwall's
inequality and the sonic boundary condition $n_1(0)=n_2(0)=1$, we get
\begin{equation}\label{27}
(n_1-n_2)^2(x)\le C\|b_1-b_2\|_{C[0, 1]}^2,\quad x\in[0,\delta_0),
\end{equation}
which in turn implies that
\begin{equation}\label{28}
|n_1-n_2|(x)+|\left(n_{1}-n_{2}\right)_x|(x)\le C\|b_1-b_2\|_{C[0, 1]},\quad x\in[0,\delta_0).
\end{equation}
with the aid of the foregoing local estimate \eqref{33} again.

Finally, from Equation \eqref{E} in Definition \ref{def}, we have
\begin{equation} \label{29}
E_i(x)=\alpha+\int_0^x\left(n_i(y)-b_i(y)\right)dy,\quad i=1,2.
\end{equation}
Taking the difference of $\eqref{29}|_{i=1}$ and $\eqref{29}|_{i=2}$, we
compute that
\begin{align}\label{30}
|E_1-E_2|(x)\le&\int_0^x|n_1-n_2|(y)dy+\int_0^x|b_1-b_2|(y)|dy\\
\le&C\|b_1-b_2\|_{C[0,1]},\quad x\in[0,\delta_0),\notag
\end{align}
and
\begin{align}\label{31}
|(E_1-E_2)_x|(x)=&|n_1-n_2-(b_1-b_2)|(x)\\
\le&C\|b_1-b_2\|_{C[0,1]}, \quad x\in[0,\delta_0).\notag
\end{align}
Hence, the local structural stability estimate \eqref{key-1} follows
immediately from Equations \eqref{28}, \eqref{30} and \eqref{31}.\qed
\vspace{3mm}

We now turn to analyzing the refined boundary behavior of the first-order
derivative of $n_i(x)$ at the right endpoint $x=1$. From the boundary estimate
 displayed in the second line of $\eqref{L14}$, we know that
 $\lim_{x\to1^-}n_{ix}(x)=-\infty$. This means the ``genuine''
 singularity will occur at the right endpoint $x=1$. Inspired
 by $\eqref{L14}$, we are able to implement the local ``weighted''
 singularity analysis. The result is summarized as follows.

\vspace{1.5mm}
\begin{lem}\label{L4-1}
Assume that $b_i, i=1,2$ satisfy the same conditions in Lemma \ref{Lcp}. Then
\begin{equation}\label{(1)}
\lim_{x\to1^-}(1-x)^{\frac{1}{2}}n_{ix}(x)=-\frac{1}{2}\sqrt{\int_0^1(b_i-n_i)dx}=:B_i<0,\quad i=1,2.
\end{equation}
\end{lem}

\vspace{3mm}
\noindent\textbf{Proof.}
For $i=1,2$, from the boundary estimate \eqref{L14}, we know that the
coefficient $1-\frac{1}{n_i^2}$ in the degenerate principal part of
Equation $\eqref{1}_1$ is comparable to $(1-x)^{\frac{1}{2}}$ near
the right endpoint $x=1$. Thus the regularity theory of
boundary-degenerate elliptic equations in one dimension
(e.g. \cite{Sc81}) ensures that $(1-x)^\frac{1}{2}n_{ix}(x)$ is
continuous up to the right endpoint $x=1$.

We now proceed to calculate the exact limit value of
$\lim_{x\to1-}(1-x)^{\frac{1}{2}}n_{ix}(x)$. For convenience, we set
\begin{equation*}
B_i:=\lim_{x\to1-}(1-x)^{\frac{1}{2}}n_{ix}(x).
\end{equation*}
Thereupon, multiplying through Equation $\eqref{1}_1$ by $(1-x)^{\frac{1}{2}}\frac{n_i^2}{n_i^2-1}$, we have
\begin{equation*}
(1-x)^\frac{1}{2}n_{ix}=\frac{n_i^3}{n_i+1}\left(E_i-\frac{\alpha}{n_i}\right)\frac{(1-x)^\frac{1}{2}}{n_i-1}.
\end{equation*}
By virtue of the sonic boundary condition $n_i(1)=1$, the known
boundary behavior \eqref{E(1)} and the L'Hospital Rule, we compute
\begin{align}\label{(3)}
B_i=&\lim_{x\to1^-}(1-x)^\frac{1}{2}n_{ix}\\
=&\lim_{x\to1^-}\frac{n_i^3}{n_i+1}\lim_{x\to1^-}(E_i-\frac{\alpha}{n_i})\lim_{x\to1^-}\frac{(1-x)^\frac{1}{2}}{n_i-1}\notag\\
=&\frac{1}{2}(E_i(1)-\alpha)\lim_{x\to1^-}\frac{-\frac{1}{2}(1-x)^{-\frac{1}{2}}}{n_{ix}}\notag\\
=&\frac{1}{4}(\alpha-E_i(1))\lim_{x\to1^-}\frac{1}{n_{ix}(1-x)^\frac{1}{2}}\notag\\
=&\frac{1}{4}(E_i(0)-E_i(1))\frac{1}{B_i},\notag
\end{align}
which implies from Equation $\eqref{1}_2$ that
\begin{equation*}
B_i=-\frac{1}{2}\sqrt{E_i(0)-E_i(1)}=-\frac{1}{2}\sqrt{\int_0^1(b_i-n_i)dx}<0,
\end{equation*}
where the boundary estimate $\eqref{L14}_2$ has been employed to uniquely
determine the value of $B_i$, which is strictly negative.\qed
\vspace{3mm}

Proposition \ref{T1} alongside Lemmas \ref{Lcp} and \ref{L4-1} now enable us
 to demonstrate the local weighted structural stability of interior subsonic
 solutions to the boundary value problem \eqref{1.6}\&\eqref{1.7} on an
 intrinsic neighborhood of the right endpoint $x=1$.

\vspace{1.5mm}
\begin{lem}[Local weighted structural stability estimate near $x=1$]\label{L4}
Under the same conditions in Lemma \ref{L1}. There exist two positive
constants $\delta_1\in(0,\frac{1}{2})$ and $C>0$ independent
of $\| b_1-b_2\|_{C[0, 1]}$ such that
\begin{align}\label{key-2}
&\left\|(1-x)^{-\frac{1}{2}}(n_1-n_2)\right\|_{C(1-\delta_1,1]}+\left\|(1-x)^\frac{1}{2}(n_{1x}-n_{2x})\right\|_{C(1-\delta_1,1]}\\
&+\|E_1-E_2\|_{C^1(1-\delta_1,1]}\le C\|b_1-b_2\|_{C[0,1]}.\notag
\end{align}
\end{lem}

\vspace{3mm}
\noindent\textbf{Proof.}
For $i=1,2$, from $\eqref{L14}_1$, we have known that $\frac{n_i-1}{(1-x)^{1/2}}$
 possesses the uniform positive upper and lower bounds near $x=1$, and
 so does its reciprocal $\frac{(1-x)^{1/2}}{n_i-1}$. This property will be
 used repeatedly hereafter.

Owing to the fact that $n_{ix}(x)$ has the genuine singularity at $x=1$, we
are compelled to establish the structural stability estimate near $x=1$ only
in the weighted manner as follows.

Firstly, multiplying through Equation $\eqref{1}_1$ by $(1-x)^{\frac{1}{2}}\frac{n_i^2}{n_i^2-1}$ and taking the difference of resultant equations for $i=1,2$, we calculate that
\begin{align}\label{(4)}
&(1-x)^\frac{1}{2}(n_{1x}-n_{2x})\\
=&\frac{n_1^2}{n_1+1}(n_1E_1-\alpha)\frac{(1-x)^\frac{1}{2}}{n_1-1}-\frac{n_2^2}{n_2+1}(n_2E_2-\alpha)\frac{(1-x)^\frac{1}{2}}{n_2-1}\notag\\
=&\frac{n_1^2}{n_1+1}(n_1E_1-\alpha)\left(\frac{(1-x)^\frac{1}{2}}{n_1-1}-\frac{(1-x)^\frac{1}{2}}{n_2-1}\right)\notag\\
&+\left(\frac{n_1^2}{n_1+1}(n_1E_1-\alpha)-\frac{n_2^2}{n_2+1}(n_2E_2-\alpha)\right)\frac{(1-x)^\frac{1}{2}}{n_2-1}\notag\\
=&h(n_1,E_1)\frac{(1-x)^\frac{1}{2}}{n_1-1}\frac{(1-x)^\frac{1}{2}}{n_2-1}\frac{n_2-n_1}{(1-x)^\frac{1}{2}}+\left(h(n_1,E_1)-h(n_2,E_2)\right)\frac{(1-x)^\frac{1}{2}}{n_2-1}\notag\\
=:&I_1+I_2,\notag
\end{align}
where
\begin{equation*}
h(n_i,E_i):=\frac{n_i^2}{n_i+1}(n_iE_i-\alpha),\quad i=1,2.
\end{equation*}

In what follows, near $x=1$, we shall estimate $I_1$ and $I_2$,
respectively. But first, we claim that the following estimates
\begin{gather}
|E_i(x)|\le\alpha+2\bar{b}_i,\quad x\in[0,1],\quad i=1,2,\label{hu1}\\
|E_1(1)-E_2(1)|\le C\|b_1-b_2\|_{C[0,1]}\label{hu2}
\end{gather}
hold, where the estimate constant $C>0$ is independent
of $\|b_1-b_2\|_{C[0,1]}$, and the proof of which is deferred
to Lemma \ref{lem-hu} at the end of this paper.

As for $I_1$, it is clear from $\eqref{L14}_1$ and \eqref{hu1} that
\begin{equation}\label{(5)}
|I_1|=\left|h(n_1,E_1)\frac{(1-x)^\frac{1}{2}}{n_1-1}\frac{(1-x)^\frac{1}{2}}{n_2-1}\frac{n_2-n_1}{(1-x)^\frac{1}{2}}\right|\le C\frac{|n_1-n_2|}{(1-x)^\frac{1}{2}}.
\end{equation}

However, as far as $I_2$ is concerned, the situation becomes more
complicated because of the factor $h(n_1,E_1)-h(n_2,E_2)$. Next, we are
taking it step by step. Precisely, a straightforward computation gives
\begin{align}\label{(6)}
h(n_1,E_1)-h(n_2,E_2)&=\frac{n_1^2}{n_1+1}(n_1E_1-\alpha)-\frac{n_2^2}{n_2+1}(n_2E_2-\alpha)\\
&=\alpha\left(\frac{n_2^2}{n_2+1}-\frac{n_1^2}{n_1+1}\right)+\left(\frac{n_1^3E_1}{n_1+1}-\frac{n_2^3E_2}{n_2+1}\right)\notag\\
&=:R_1+R_2.\notag
\end{align}
From \eqref{ulb} and Lemma \ref{Lcp}, we know that
\begin{equation}\label{hu0}
1\le1+m(\alpha,\underline{b}_2)\sin(\pi x)\le n_2(x)\le n_1(x)\le\overline{b}_1,\quad x\in[0,1].
\end{equation}
Consequently, it follows from the mean-value theorem of differentials that
\begin{equation}\label{(7)}
|R_1|=\left|\alpha\left(\frac{n_2^2}{n_2+1}-\frac{n_1^2}{n_1+1}\right)\right| \le C|n_1-n_2|\le C\frac{|n_1-n_2|(x)}{(1-x)^\frac{1}{2}}.
\end{equation}
We now turn to estimating $R_2$ near $x=1$. Combining \eqref{hu0},
\eqref{hu1}, \eqref{hu2}, the mean-value theorem of differentials,
and the mean-value theorem of integrals, we have
\begin{align}\label{hu3}
|R_2|&=\left|\frac{n_1^3E_1}{n_1+1}-\frac{n_2^3E_2}{n_2+1}\right|\\
&=\left|E_1\left(\frac{n_1^3}{n_1+1}-\frac{n_2^3}{n_2+1}\right)+\frac{n_2^3}{n_2+1}(E_1-E_2)\right|\notag\\
&\le C|n_1-n_2|(x)+C\left|\left[\big(E_1(1)-E_2(1)\big)-\left(\int_x^1(n_1-n_2)-(b_1-b_2)dy\right)\right]\right|\notag\\
&\le C|n_1-n_2|(x)+C|E_1(1)-E_2(1)|+C\int_x^1|n_1-n_2|dy+C\|b_1-b_2\|_{C[0,1]}\notag\\
&\le C\|b_1-b_2\|_{C[0,1]}+C\left(\frac{|n_1-n_2|(x)}{(1-x)^{\frac{1}{2}}}+\frac{|n_1-n_2|(\xi)}{(1-\xi)^{\frac{1}{2}}}\right),\quad\exists\xi\in[x,1],\notag
\end{align}
where we have used the formula
\begin{equation}
E_i(x)=E_i(1)-\int_x^1(n_i-b_i)(y)dy,\quad i=1,2.
\end{equation}
Substituting \eqref{(7)} and \eqref{hu3} into \eqref{(6)}, we have
\begin{equation*}
|h(n_1,E_1)-h(n_2,E_2)|\le C\|b_1-b_2\|_{C[0,1]}+C\left(\frac{|n_1-n_2|(x)}{(1-x)^{\frac{1}{2}}}+\frac{|n_1-n_2|(\xi)}{(1-\xi)^{\frac{1}{2}}}\right),\quad\exists\xi\in[x,1],
\end{equation*}
which further implies that
\begin{align}\label{hu4}
|I_2|&=\left|\left(h(n_1,E_1)-h(n_2,E_2)\right)\frac{(1-x)^\frac{1}{2}}{n_2-1}\right|\\
&\le C\|b_1-b_2\|_{C[0,1]}+C\left(\frac{|n_1-n_2|(x)}{(1-x)^{\frac{1}{2}}}+\frac{|n_1-n_2|(\xi)}{(1-\xi)^{\frac{1}{2}}}\right),\quad\exists\xi\in[x,1].\notag
\end{align}

Inserting \eqref{(5)} and \eqref{hu4} into \eqref{(4)}, near $x=1$, we
obtain \begin{align}\label{(14)}
&(1-x)^\frac{1}{2}|(n_1-n_2)_x|(x)\\
\le&C\left(\frac{|n_1-n_2|(x)}{(1-x)^{\frac{1}{2}}}+\frac{|n_1-n_2|(\xi)}{(1-\xi)^{\frac{1}{2}}}\right)+C\|b_1-b_2\|_{C[0,1]},\quad\exists\xi\in[x,1].\notag
\end{align}
It is worth mentioning that the generic constant $C>0$ in \eqref{(14)} is
independent of $\|b_1-b_2\|_{C[0,1]}$. Moreover,
the term $\frac{|n_1-n_2|(x)}{(1-x)^{1/2}}+\frac{|n_1-n_2|(\xi)}{(1-\xi)^{1/2}}$
on the right-hand side of \eqref{(14)} can be bounded by an appropriate
constant multiple of $\|b_1-b_2\|_{C[0,1]}$ in an intrinsic neighborhood
of the right endpoint $x=1$. Precisely, we claim that there exist two
positive constants $0<\delta_1<\frac{1}{2}$ and $M_1>0$ independent
of $\|b_1-b_2\|_{C[0,1]}$ such that
\begin{equation}\label{(14*)}
\frac{|n_1-n_2|(x)}{(1-x)^\frac{1}{2}}\le M_1\|b_1-b_2\|_{C[0,1]}, \quad x\in(1-\delta_1,1].
\end{equation}
Aiming for a contradiction, suppose that for any $\delta\in (0,\frac{1}{2})$
and $M>0$, there is $x_\delta\in(1-\delta,1]$ such that
\begin{equation}
\frac{|n_1-n_2|(x_\delta)}{(1-x_\delta)^\frac{1}{2}}>M\|b_1-b_2\|_{C[0,1]}.
\end{equation}
By the arbitrariness, we could take $\delta=\frac{1}{k}, k=3,4,5,\cdots$, for
arbitrary $M>0$, there exists $x_k\in(1-\frac{1}{k},1]$ such that
\begin{equation}
\frac{|n_1-n_2|(x_k)}{(1-x_k)^\frac{1}{2}}>M\|b_1-b_2\|_{C[0,1]},
\end{equation}
which implies that
\begin{equation}\label{0227-5}
\varliminf_{x_k\to1^-}\frac{|n_1-n_2|(x_k)}{(1-x_k)^\frac{1}{2}}\ge M\|b_1-b_2\|_{C[0,1]}.
\end{equation}
Besides, combining Lemma \ref{Lcp}, the L'Hospital Rule and Lemma
\ref{L4-1}, we calculate that
\begin{align}\label{0227-6}
\lim_{x\to1^-}\frac{|n_1-n_2|(x)}{(1-x)^\frac{1}{2}}=&\lim_{x\to1^-}\frac{(n_1-n_2)(x)}{(1-x)^\frac{1}{2}}=\lim_{x\to1^-}\frac{(n_1-1)(x)}{(1-x)^\frac{1}{2}}-\lim_{x\to1^-}\frac{(n_2-1)(x)}{(1-x)^\frac{1}{2}}\\
=&-2\lim_{x\to1^-}(1-x)^{\frac{1}{2}}n_{1x}+2\lim_{x\to1^-}(1-x)^{\frac{1}{2}}n_{2x}\notag\\
=&\sqrt{\int_0^1(b_1-n_1)dx}-\sqrt{\int_0^1(b_2-n_2)dx}\notag\\
=&\frac{\int_0^1(b_1-b_2)dx-\int_0^1(n_1-n_2)dx}{\sqrt{\int_0^1(b_1-n_1)dx}+\sqrt{\int_0^1(b_2-n_2)dx}}\notag\\
\le&\frac{\int_0^1(b_1-b_2)dx}{\sqrt{\int_0^1(b_1-n_1)dx}+\sqrt{\int_0^1(b_2-n_2)dx}}\notag\\
\le&\tilde{C}_1\|b_1-b_2\|_{C[0,1]}.\notag
\end{align}
Moreover, we note that the constant $M>0$ in \eqref{0227-5} is arbitrary.
Therefore, together with \eqref{0227-6}, taking $M=2\tilde{C}_1$
in \eqref{0227-5} leads to the contradiction that $2\le1$.

Applying \eqref{(14*)} to \eqref{(14)}, we have
\begin{equation}\label{hu5}
(1-x)^\frac{1}{2}|(n_1-n_2)_x|(x)\le C\|b_1-b_2\|_{C[0,1]}, \quad x\in(1-\delta_1,1].
\end{equation}

Similarly to \eqref{hu3}, we are able to compute that
\begin{align}\label{(21)}
|E_1-E_2|(x)\le&C\|b_1-b_2\|_{C[0,1]}+\frac{|n_1-n_2|(\xi)}{(1-\xi)^{\frac{1}{2}}},\quad\exists\xi\in[x,1]\\
\le&C\|b_1-b_2\|_{C[0,1]},\quad x\in(1-\delta_1,1],\notag
\end{align}
and
\begin{align}\label{(22)}
|(E_1-E_2)_x|(x)=&|n_1-n_2-(b_1-b_2)|(x)\le|n_1-n_2|(x)+|b_1-b_2|(x)\\
\le&\frac{|n_1-n_2|(x)}{(1-x)^{\frac{1}{2}}}+\|b_1-b_2\|_{C[0,1]}\notag\\
\le&C\|b_1-b_2\|_{C[0,1]},\quad x\in(1-\delta_1,1].\notag
\end{align}

Finally, putting results \eqref{(14*)}, \eqref{hu5}, \eqref{(21)}
and \eqref{(22)} together, we obtain the desired local weighted
estimate \eqref{key-2}.\qed
\vspace{3mm}

Up to now, we have obtained two intrinsic small domains $[0,\delta_0)$
and $(1-\delta_1,1]$ distributed around the two endpoints $x=0$ and $x=1$,
respectively. This fact enables us to establish the structural stability
estimate on a certain regular domain $[\delta,1-\delta]$, where
$0<\delta:=\min\{\delta_0,\delta_1\}<1/2$.

\vspace{1.5mm}
\begin{lem}\label{L5}
Under the same conditions in Lemma \ref{L1}. Let $\delta:=\min\{\delta_0,\delta_1\}$. Then there is a positive constant $C>0$ independent of $\|b_1-b_2\|_{C[0,1]}$ such that
\begin{equation}\label{key-3}
\|n_1-n_2\|_{C^1[\delta,1-\delta]}+\|E_1-E_2\|_{C^1[\delta,1-\delta]}\le C\|b_1-b_2\|_{C[0,1]}.
\end{equation}
\end{lem}

\vspace{3mm}
\noindent\textbf{Proof.}
We are now able to work with Equations \eqref{1} on a regular closed
interval $[\delta,1-\delta]$ away from singularities, where $\delta$
has been defined in the hypothesis of the present lemma.

Firstly, we rewrite the estimate \eqref{hu0} on the regular interval as follows.
\begin{equation}\label{hu6}
1<l:=1+m(\alpha,\underline{b}_2)\sin(\pi\delta)\le n_2(x)\le n_1(x)\le\overline{b}_1,\quad x\in[\delta,1-\delta].
\end{equation}

Secondly, subtracting $\eqref{1}|_{i=2}$ from $\eqref{1}|_{i=1}$, for $x\in[\delta,1-\delta]$, we thus get
\begin{align}\label{41}
(n_1-n_2)_x=&\frac{n_1^3E_1-\alpha n_1^2}{n_1^2-1}-\frac{n_2^3E_2-\alpha n_2^2}{n_2^2-1}\\
=&E_1\left(f(n_1)-f(n_2)\right)+f(n_2)(E_1-E_2)-\alpha\left(g(n_1)-g(n_2)\right)\notag\\
=&\left(E_1 f'(\bar{\eta})-\alpha g'(\tilde{\eta})\right)(n_1-n_2)+f(n_2)(E_1-E_2),\quad\exists\bar{\eta},\tilde{\eta}\in(n_2,n_1),\notag
\end{align}
and
\begin{equation}\label{42}
(E_1-E_2)_x =(n _1-n _2)-(b_1-b_2),
\end{equation}
where
\begin{equation*}
f(n):=\frac{n^3}{n^2-1},\quad g(n):=\frac{n^2}{n^2-1}, \quad\forall n\in[l,\overline{b}_1],
\end{equation*}
and we have used the mean-value theorem of differentials in the third
line of Equation \eqref{41}.

Thirdly, multiplying through \eqref{41} by $n_1-n_2$, and using \eqref{hu1},
\eqref{hu6} and Cauchy's inequality together, we have
\begin{equation}\label{hu7}
\left((n_1-n_2)^2\right)_x\le C(\alpha,l,\bar{b}_1)\left((n_1-n_2)^2+(E_1-E_2)^2\right), \quad x\in[\delta,1-\delta].
\end{equation}
Similarly, multiplying through \eqref{42} by $E_1-E_2$, and employing
Cauchy's inequality, we obtain
\begin{equation}\label{hu8}
\left((E_1-E_2)^2\right)_x\le(n_1-n_2)^2+2(E_1-E_2)^2+\|b_1-b_2\|_{C[0,1]}^2, \quad x\in[\delta,1-\delta].
\end{equation}
And then, summing estimates \eqref{hu7} and \eqref{hu8} gives
\begin{align}\label{hu9}
&\left((n_1-n_2)^2+(E_1-E_2)^2\right)_x(x)\\
\le&C\left((n_1-n_2)^2+(E_1-E_2)^2\right)(x)+\|b_1-b_2\|_{C[0,1]}^2, \quad x\in[\delta,1-\delta].\notag
\end{align}
Applying the Gronwall inequality to \eqref{hu9}, we have
\begin{align}\label{hu10}
&\left((n_1-n_2)^2+(E_1-E_2)^2\right)(x)\\
\le&e^{\int_\delta^xCdy}\left[\left((n_1-n_2)^2+(E_1-E_2)^2\right)(\delta)+\int_\delta^x\|b_1-b_2\|_{C[0,1]}^2dy\right]\notag\\
\le&C\left[\left((n_1-n_2)^2+(E_1-E_2)^2\right)(\delta)+\|b_1-b_2\|_{C[0,1]}^2\right], \quad x\in[\delta,1-\delta].\notag
\end{align}
Noting that $\delta\le\delta_0$ and the continuity of the error
function pair $(n_1-n_2, E_1-E_2)(x)$ at $x=\delta_0$, from
Lemma \ref{L3} we see
\begin{equation}\label{hu11}
\left((n_1-n_2)^2+(E_1-E_2)^2\right)(\delta)\le C\|b_1-b_2\|_{C[0,1]}^2,
\end{equation}
which along with \eqref{hu10} implies
\begin{equation}\label{47}
|n_1-n_2|(x)+|E_1-E_2|(x)\le C\|b_1-b_2\|_{C[0,1]}, \quad x\in[\delta,1-\delta].
\end{equation}

Finally, from Equations \eqref{41}\&\eqref{42}, and the estimate \eqref{47},
 we directly calculate
\begin{equation}\label{hu12}
|(n_1-n_2)_x|(x)+|(E_1-E_2)_x|(x)\le C\|b_1-b_2\|_{C[0,1]}, \quad x\in[\delta,1-\delta].
\end{equation}
Combining estimates \eqref{47} and \eqref{hu12} yields the desired structural
 stability estimate \eqref{key-3} on the regular domain $[\delta,1-\delta]$.\qed
\vspace{3mm}

Last but not least, let us prove the estimates \eqref{hu1} and \eqref{hu2} in
 the following lemma.

\vspace{1.5mm}
\begin{lem}\label{lem-hu}
Under the same conditions in Lemma \ref{L4}. Then there exists a
positive constant $C$ independent of $\|b_1-b_2\|_{C[0,1]}$ such that
 estimates \eqref{hu1} and \eqref{hu2} hold, that is,
\begin{equation*}
|E_i(x)|\le\alpha+2\bar{b}_i,\quad x\in[0,1],\quad i=1,2,
\end{equation*}
and
\begin{equation*}
|E_1(1)-E_2(1)|\le C\|b_1-b_2\|_{C[0,1]},
\end{equation*}
respectively.
\end{lem}

\vspace{3mm}
\noindent\textbf{Proof.}
From Equation \eqref{E} in Definition \ref{def}, we have
\begin{equation} \label{hu13}
E_i(x)=\alpha+\int_0^x(n_i-b_i)(y)dy,\quad\forall x\in[0,1], \quad i=1,2.
\end{equation}

First of all, in light of the lower and upper bounds \eqref{ulb} of $n_i(x)$,
a straightforward computation gives
\begin{align}
|E_i(x)|=\left|\alpha+\int_0^x(n_i-b_i)dy\right|\le\alpha+\int_0^1(n_i+b_i)dy\le\alpha+2\bar{b}_i,\quad\forall x\in[0,1], \quad i=1,2.
\end{align}

Next, taking the value $x=1$ in Equation \eqref{hu13}, we have
\begin{equation}\label{hu14}
E_i(1)=\alpha+\int_0^1(n_i-b_i)(y)dy, \quad i=1,2.
\end{equation}
Furthermore, taking the difference of Equations $\eqref{hu14}|_{i=1}$
and $\eqref{hu14}|_{i=2}$, we calculate
\begin{align}\label{hu15}
|E_1(1)-E_2(1)|\le&\left|\int_0^1\Big[(n_1-b_1)-(n_2-b_2)\Big]dy\right|\\
\le&\int_0^1|n_1-n_2|(y)dy+\|b_1-b_2\|_{C[0,1]}\notag\\
=&|n_1-n_2|(\xi)+\|b_1-b_2\|_{C[0,1]}, \quad\exists\xi\in[0,1],\notag
\end{align}
where we have used the mean-value theorem of integrals in the last line.

Finally, we claim that there is a positive constant $C$ independent of
$\|b_1-b_2\|_{C[0,1]}$ such that
\begin{equation}\label{hu16}
|n_1-n_2|(\xi)\le C\|b_1-b_2\|_{C[0,1]},
\end{equation}
wherever the point $\xi$ is located in the whole interval $[0,1]$.
In fact, take $\delta$ the same as in Lemma \ref{L5}, and if
$\xi\in[0,1-\delta]$, it is clear from estimates \eqref{key-1}
and \eqref{key-3} that \eqref{hu16} is true; if $\xi\in(1-\delta,1]$,
the intrinsic local estimate \eqref{(14*)} guarantees that \eqref{hu16}
is true as well. Consequently, substituting \eqref{hu16} into \eqref{hu15},
we obtain the desired estimate \eqref{hu2}.\qed
\vspace{3mm}

We end this section by a summary for the proof of Theorem \ref{T2}
because the previous lemmas have already made the proof evident.

\vspace{3mm}
\noindent\textbf{Proof of Theorem \ref{T2}.}
Obviously, putting all the estimates we have established in
Lemmas \ref{L3}, \ref{L5} and \ref{L4} together, we have the
globally structural stability estimate \eqref{1.10**}.\qed

\vspace{5mm}
\noindent{\sc Acknowledgments:} This work was commenced while the
first two authors were visiting McGill University from 2021 to 2022.
They would like to express their gratitude to McGill University for
its hospitality. The research of Y. H. Feng was supported by China
Scholarship Council for the senior visiting scholar program
(202006545001). The research of H. Hu was partially supported by
National Natural Science Foundation of China (Grant No.11801039),
China Scholarship Council (No.202007535001) and Scientific Research
Program of Changchun University (No.ZKP202013). The research of M.
Mei was partially supported by NSERC grant RGPIN 354724-2016.

redpajama_set_name: RedPajamaArXiv


text:
STAR PEAK 6167' TRAIL #998
STAR PEAK 6167'. TRAIL #998
EVENT TYPE: Day hiking, backpacking, equestrian
DISTANCE: about 8 miles RT
ELEVATION GAIN: 4116'
DIFFICULTY: Strenuous
MAPS: Kootenai N.F., Heron topo
RANGER DISTRICT: Cabinet R.D. 406.827.3533
SANDERS COUNTY SHERIFF: 406.827.3584
Trail #998 starts near the Big Eddy Campground and Boat launch along the Clark Fork River, on Highway 200. The actual trailhead was relocated in 2016-7.
Most of the trail is up thru thick forests of Douglas Fir, until higher up the trail, where Lodgepole pine takes over. Be sure to stop and rest occasionally along this trail.
As you approach the summit, the views are restricted to the trees. But as you walk onto the summit block, the views open up.
In the middle of the summit area stands an old log cabin that was used for storage for the lookout personnel. The active lookout tower remains on it's rocky pinnacle above to the east. Take a moment to walk to it for some of the most spectacular views in the area.
But there's something else that draws people to this peak. Off in the distance to the south rim of the summit, is the most spectacular outhouse you can imagine. It sits on the very edge of the summit block. From near the outhouse, you can look down some 50+ degrees to the Clark Fork River and Hwy 200..
Take extra care descending this trail. It is steep in places and hard on the knees.
From Clark Fork, head east on Highway 200 into Montana for 6 miles and look for a Big Eddy sign along Hwy 200. Look for the new parking area. Across from the Big Eddy sign is a steep road that leads to the trailhead.
Long hard up with lots of elevation gain. Take many rest breaks on this hike.
The Clark Fork River, Cabinet Gorge Dam, Heron, Pillick Ridge, the Cabinet Mountain Wilderness.
Clark Fork Pantry, Clark Fork Jalapeños, Mr. Sub, Burger Express, and Eichardt's in Sandpoint
On the way up looking west towards Clark Fork
IMAGE BY DAVID CRAFTON
TRAIL JUNCTION WITH BIG EDDY TRAIL
LOOKING NORTH FROM THE SUMMIT
IMAGE BY DAVID CRAFTON
THE STONE HOUSE BELOW THE LOOKOUT TOWER.
THIS HOUSE USES ZERO MORTOR IN ITS CONSTRUCTION.
IMAGE BY CHRIS H.
BILLIARD TABLE PEAK 6622' LOOKING NORTH FROM STAR PEAK 6167'
THE RIDGE BETWEEN STAR PEAK AND BILLIARD TABLE MOUNTAIN
IMAGE BY CHRIS H.
ALONG THE ROUTE TO BILLIARD TABLE MOUNTAIN
THE MAIN ATTRACTION AT STAR PEAK. THIS OUTHOUSE HAS AN INCREDIBLE VIEW.
A REAL POOP WITH A VIEW.
THE HISTORY OF THE SQAW PEAK 6167' (NOW STAR PEAK)
1906: Established as an unimproved observation point.
December 27, 1907: "The supervisor expects some time this winter or early in the spring to begin building the telephone line up Bull river to the lookout point on Squaw peak." (The Sanders County Independent-Ledger)
June 19, 1908: "The trail from Bull river to the top of Squaw peak, where a lookout station will be maintained during the fire season, was completed recently and is now open from the mouth of Star gulch to the summit." (The Sanders County Ledger)
July 23, 1909: "The Squaw Peak lookout station, at the west end of the Cabinet National Forest, in charge of Forest Guard James Berray, has already shown its efficiency by reporting the discovery of three fires in the Coeur d'Alene forest and one in the valley near Noxon, outside of the Cabinet forest boundaries, the presence of none of which had not become known to anybody prior to the lookout reporting them. A telephone line from the peak affords communication with the supervisor's headquarters and the several ranger stations and no sooner is a fire discovered by the lookout than are steps on foot for its quelling. The Mt. Silcox station will soon be occupied. At present the mountain top is the abiding place, day and night, for three forest service men engaged in trail work, and they keep a lookout for fires, though hereabout there is but little damage just yet, unless a careless camper leaves his fire unguarded." (The Sanders County Ledger)
July 27, 1909: "It has been reported to the district forester that forest fires are now burning in national forests in Idaho, and that, incidentally, the practicability of lookout stations has been demonstrated. One fire is on Gold creek, north of Clark's Fork, Idaho, and is said to be doing considerable damage. It was discovered by a forest guard stationed in a lookout cabin on the top of Squaw peak, a jagged mountain of the locality which rears its head far above the surrounding country and enables anyone on its summit to overlook the country for 65 miles around. From this point one can see from Sandpoint to Trout Creek. The station is fitted with telephone facilities, and when the fire broke out the supervisor, 75 miles away, was promptly notified and got assistance to the scene in short time. A large force of rangers and guards is now fighting the flames." (The Daily Missoulian)
October 15, 1909: "Supervisor R.H. Bushnell devoted all last week to inspecting permanent improvement work and looking over timber sales in the Bull river and Noxon district. He visited Squaw peak and declares it an ideal lookout station during the fire season. He says the entire Clarksfork valley from Sandpoint on the west to the Blue Slide on the east, is easily visible from its top and it also affords a splendid view of the Elk creek, Pilgrim creek and Bull river valleys. Mrs. Bushnell accompanied him on his journey on horse-back." (The Sanders County Ledger)
1910: A stone cabin was built to serve as living quarters for the lookout. A pillar of rocks piled for use as the observation post.
July 12, 1912: "Sam Shafer has accepted a position as lookout on Squaw peak for the forest service." (The Sanders County Ledger)
September 24, 1915: "Mr. Hudson a student at Missoula who has been lookout man on Squaw peak, left here Wednesday." (The Sanders County Ledger)
July 15, 1920: "Ranger Saint has installed the Squaw Peak and Gem Peak lookouts during the last few days." (The Sanders County Independent-Ledger)
September 24, 1930: "The work of installing four ready cut lookout houses has been finished. The buildings were put on Squaw Peak in the Noxon district, Seven Point Peak in the Trout Creek district, and at Big Hole and Penrose Peaks in the Plains district." (The Sanders County Independent-Ledger)
1930: A R-1 style gable roof ground house was constructed.
August 31, 1950: "Jim Brooks had been a lookout on Squaw peak and had just been released that morning due to the heavy rains." (Sandpoint News Bulletin)
1952: A L-4 ground cabin. (Kresek)
June 7, 1956: "Jim Dewey, assistant ranger, announced plans to start construction of the Squaw Peak lookout tower about June 4." (Sandpoint News-Bulletin)
July 5, 1956: "The forest service lookout crews were sent out June 25 and 26. Mr. and Mrs. Earl Robinette from Nebraska are on Squaw peak." (Sandpoint News-Bulletin)
September 6, 1956: "The forest service has released all its lookout crews for the season Dean Henderson, who was on Green mountain lookout, went up on Squaw peak Tuesday to man the automatic repeat station there until Sept. 20. Velmer Munsen accompanied him." (Sandpoint News-Bulletin)
June 27, 1957: "Dean Henderson and Velmer Munsen left Monday for Squaw peak lookout station. Dean will return this week end and then go to Green mountain peak lookout." (Sandpoint News-Bulletin)
September 13, 1962: "David Runkle was sent up to Squaw Peak the first of last week to replace the lookout who came down a few days before returning to school." (Sanders County Ledger)
July 4, 1963: "Mr. and Mrs. David Runkle went up on Squaw Peak Monday where they will spend the summer." (Sanders County Ledger)
August 1, 1963: "Mr. and Mrs. David Runkle, who are stationed at the Squaw Peak lookout were flown by helicopter to Sandpoint Friday when Mrs. Runkle needed medical attention following a spider bite. They spent the remainder of the weekend with relatives here and returned to the lookout Tuesday." (Sanders County Ledger)
September 19, 1963: "Mr. and Mrs. David Runkle came down from Squaw Peak Tuesday. They had spent the summer as lookouts there. They motored to Spokane Friday when David registered for school. They returned to Spokane again Sunday as classes at the Spokane Trade school started Monday." (Sanders County Ledger)
September 22, 1966: "Miss Laura Harker came down from Squaw Peak Thursday. She has been lookout there this summer. She was down over labor day but had to go back up due to the weather conditions." (Sandpoint News Bulletin)
April 30, 1967: "Miss Harker, who spent three months on Squaw Peak without a break, had only 50 visitors all summer. She had n o electricity and used a wood stove for heat but she says she is ready for another year." (The Independent Record)
October 12, 1967: "Due to the rains last weekend, Laura Harker came down from Squaw Peak on Monday when the lookout was closed. She had spent the summer up there." (Sandpoint News Bulletin)
September 10, 1970: "Mary Ballenger of Squaw Peak Lookout will remain on Lookout until her duties are finished." (Sandpoint News Bulletin)
2004: The mountain suffered a name change from the historic Squaw Mountain to the present Star Peak.

redpajama_set_name: RedPajamaCommonCrawl


Below - an article from the Spring 1997 edition of the former Poole Pottery Collectors Club magazine. The magazines are an excellent resource and most editions are still available. The fake cherubs which prompted the article turn up regularly on ebay and at auction houses.
A selection of fake marks is shown below. Most are unlikely to fool collectors if they are familiar with the genuine marks. Often the unlikely nature of the piece is the biggest giveaway that the mark is not genuine.


As a team we reviewed the organisation's mission, vision and values as well as sharing ideas to improve our services. Everyone had a lot to contribute and it was a positive morning.
We were introduced to mindfulness before lunch, then ended the day with a fun team building activity (we escaped)!


text:
The Courtyard of…
The Courtyard of the Lions receives an award from the Andalusia Marble Federation
Tonight at the 27th Edition of the Macael Awards (Premios Macael) the Court of the Lions of the Alhambra...
Tonight at the 27th Edition of the Macael Awards (Premios Macael) the Court of the Lions of the Alhambra will receive the National Prize for the "restoration work performed to restore the Court to its original condition by paving it with 250 pieces of Macael White Marble".
The Director of the Council of the Alhambra and Generalife, María del Mar Villafranca, will receive the prize, which is awarded in recognition of the work carried out by a multidisciplinary team made in Andalusia, made up of over 100 professionals who work for 33 different companies. In 2012 over 2,000,000 people visited this emblematic space in the heart of the Alhambra.
With these prizes the Andalusia Marble Federation (AEMA) pays tribute to those individuals and institutions that have supported the Marble industry. In addition to the Alhambra, AEMA also awarded prizes to the journalist, Carlos Herrera, who received the Communicator Award, and the motorcyclist, Jorge Lorenzo, who received the Person Award for his charity work.
In this year's edition of the awards, the main new feature was the numerous international awards for building works using Macael Marble in the different continents of the world. The award for North America went to the architect Daniel Liebskind; for South America to the Mexican company Luq Stones; for Africa the award was for the Misterios del Rosario monument in Equatorial Guinea and for Europe the award went to the So Ouest 4 Estrellas shopping centre in Paris.


text:
studio jobba
DesignLab Interior Design
The design of a series of furnitures for the DesignLab at the campus of the University of Twente
The University of Twente DesignLab is a multidisciplinary working space of 2000m2 located at its campus.
In the summer of 2014, two colleague students and me were assigned to design and build an interior from scratch that divides the big space into smaller ones, while still connecting the space as a whole. This had to take place in a time slot of 7 weeks.
At first, a general style is formed to ensure that every furniture has the same form language, and so bind them together. Collages and sketches are made and markings on the floor are taped in order to find the right dimensions. Final designs are 3D-modelled and either self-built or outsourced to builders, furnishers and interior building company Brok. Green is the colour that runs through the entire DesignLab; a colour that embodies life, freshness and creativity.
Sketches by Ruben van den Hout
'Low poly' is the style that is chosen, a style which uses simple edged shapes to define the total shape.
The 'faced' / 'edged'-designs create a coherence between the furnitures in the series. All designs are done by Bram Norp, Ruben van den Hout (at the time fellow students industrial design at the University of Twente) and me. Except from 'Lectern', which was done solely by me. A selection of the realised designs is shown below.
Lectern was designed for one of the lecture halls in the DesignLab and was built by Brok Interieurbouw B.V. The two-leveled desktop hides any item that is stalled out from the audience .The angle on the presenter's side allows him/her to move towards and away from the lectern in a smooth and open manner. Lockable drawers enable for laptop storage and are flushed with the dekstop, hiding it.
Family Table was desiged as a connection between the ideation, break and conceptualization space in the DesignLab. It is built by Brok Interieurbouw B.V. The main idea behind this table is that it allows for multiple people to sit at the table while still being separated by means of the different levels. Transparent legs create the illusion of a floating surface. One of the angles doubles as a magazine storage. The different levels play with the human imagination, and allow people to position themselves at a level that they are most comfortable at.
Special thanks to Jorien Alers, Ruud van Leeuwen and Niek van den Hout for their coordination throughout the project.
2020, Job van Dongen


This natural Platinum Nugget from Kondyer Massif in the Eastern Siberian Region of Russia is extremely rare. Naturally layered with gold, it is a geological phenomenon. It is very rare to see platinum nuggets like this and Siberia is the only place that they have been found. This amazing gold layered platinum specimen is 3/16" (5 mm) long, and weighs 7.7 Grains (0.50 Gram)(2.5 Carats). It comes with a location card authenticating the discovery location and certifying the fact that it is a natural platinum crystal nugget.


You don't have to be able to see to crochet. You can feel your way through the work. In fact, many people who crochet are blind – either because they were born that way or because they experience macular degeneration over time. Many older people with limited eyesight can no longer drive or do other daily things but they can still crochet. Here are some of those inspiring crochet stories about blind crocheters.
I wrote about Patricia Belcher, legally blind since birth, back in 2012 when she was aspiring to break a crochet world record for the World's Longest Crochet Chain. To my knowledge she has not yet broken this record.
Terry shared her story with us previously about crocheting to heal through the depression and fear of unemployment. She had read Em's story in my book, Crochet Saved My Life, and it had resonated for her. She mentioned in passing as she shared her story that she is legally blind. Another story that's also in Crochet Saved My Life is from Liza who crocheted to cope with temporary blindness.
Beverly Lewis was blind when she learned to crochet. Someone at church held her hands and showed her the basic stitches. She crocheted a blanket for her pastor, which took almost a year to do, but then went on to crochet presents for all fifteen of her siblings!
Talking Crochet shared that fashion designer Stephanie Dunn was "born with macular degeneration, a progressive eye disease, as well as high myopia, but (also had) an injury so damaged an optic nerve that she sees only light and shadows with her right eye." She creates knit fashions with crochet trims and details.
Gina Pontelandolfo is one of the contributors to the crochet coral reef project in Melbourne. She has a rare genetic condition "and since the age of 25 has been slowly going blind as her vision has narrowed into an ever narrower tunnel".
Kay Foster started slowly going blind about fifteen years ago and that was when she decided to learn how to crochet. She crafts blankets for charity, and she says it relaxes her to know she's helping someone else this way. She crochets 3-4 hours per day and has donated about 250 blankets. Jeanne Gallegos has a similar story.
Anna Murphy, slowly going blind from macular degeneration, reduces the anxiety that causes her by crocheting half a dozen blankets per month for charity. Similarly, Bea White crochets lap robes for cancer patients and Eva Lilly crochets prayer shawls for cancer patients. Terra Taylor is a young grandmother of three who also crochets for charity and also has macular degeneration (although she actually first lost her vision in one eye due to a freak accident).


A closed, insulated and heated mixing chamber, in which a mixture is heated with dry steam by a shaft which can rotate in two directions. Generously sized paddles, adjustable in both height and pitch, create a homogeneous mixture and a hygienic process. Two PT 100 temperature sensors register the preset temperature after which the electro-pneumatically controlled flap of the outlet is opened.
2 temperature sensors PT 100.
Almex Extrusion Techniques will be present at the 86th International Agricultural Fair in Serbia.


text:
import Ember from 'ember';

export default Ember.TextArea.extend({
  keyDown: function(event) {
    if (this._isValidCombination(event)) {
      event.preventDefault();
      this.sendAction('modifiedSubmit');
    }
  },

  _hasCorrectModifier: function(event) {
    return event.ctrlKey || event.metaKey;
  },

  _isCorrectKeyCode: function(keyCode) {
    return keyCode === 13;
  },

  _isValidCombination: function(event) {
    return this._hasCorrectModifier(event) && this._isCorrectKeyCode(event.keyCode);
  }
});

redpajama_set_name: RedPajamaGithub


text:
Warwick Hills Golf and Country Club is a private country club located at 9057  South Saginaw Road in Grand Blanc Township, Michigan, a suburb of Flint, Michigan, United States. From 1958 to 2009, Warwick Hills was home to the PGA Tour's Buick Open tournament. From 2018, it is host to the Ally Challenge event on the PGA Tour Champions. The course was opened in June 1957. The country club also participates in a caddie program.

The Warwick Hills golf course plays to a par of 72 with a length of 7112 from the championship tees. Tiger Woods and Billy Mayfair jointly hold the course record of 61.

That tournament ended after 2009 because of the government seizure of General Motors leading to cessation of golf sponsorship, thus ending the PGA Tour's association with Warwick Hills.  The Ally Challenge was first played at the course in 2018.

In 2009, a Michigan Historical Marker about the Buick Open was erected at the course.

References

External links
Official website

Golf clubs and courses in Michigan
Sports in Flint, Michigan
Buildings and structures in Genesee County, Michigan
Sports venues completed in 1957

redpajama_set_name: RedPajamaWikipedia


The overwhelming majority of the population of the U.S. is against being dragged into another disastrous war. Nothing is more dangerous than the aggressive U.S./NATO troop movements right on the borders of Russia.
Sending U.S. destroyers into the Black Sea and the Baltic Sea; scheduling threatening U.S./NATO war games and troop movements in East Europe; and imposing sanctions on the Russian Federation is a threat to peace on a world scale. We have seen the cost of past and continuing U.S. wars, which enrich the military corporations while impoverishing the targeted countries as well as poor and working people here in the U.S.
The years of U.S. funding of fascist forces in Ukraine and the recognition of a government in Kiev that overthrew the elected government, seized power and appointed extreme right-wing groups to head the police, army and national guard in order to pull Ukraine into NATO membership makes the U.S. complicit in the complete denial of the rights of the Ukrainian people. It is also a provocation against the entire region.
People in East and South Ukraine, outraged by this coup government, have attempted to resist the illegal junta, have declared an independent People's Republic of Donetsk, and have called for referendums. In response, the right-wing coup government has allowed its military forces and other fascists to terrorize the Ukrainian people. In the most recent incident, some 40 people were massacred in the city of Odessa on May 2 by fascist militants, loyal to the Kiev government, who set the Trades Union Building on fire. In addition, 23 people were killed at Slavyansk and in Kramatorsk in the Donetsk region in attacks by Ukrainian military forces from May 2-3.
Despite mass desertions by Ukrainian police and military personnel, so-called "anti-terrorist" campaigns against activists in southeastern Ukraine were launched immediately after visits to Kiev by U.S. officials. Washington has spent $5 billion to effect "regime change" in Ukraine, helping to bring into power a junta dominated by fascist, racist, anti-Semitic organizations like Svoboda, Fatherland and Right Sector. Meanwhile, the U.S. has pledged up to $10 billion in loans to the illegal coup regime, and Washington has been instrumental in securing a $17 billion aid and austerity package from the International Monetary Fund.
3. That U.S. military forces immediately be withdrawn from the Eastern European region and that NATO's expansion and provocative actions against Russia be ended.
Tragically, neither the US nor the EU honored the February 21 comprmise accord between the Maidan coalition and the Yanukovich govenment that was brokered by Foreign Ministers of France, Germany and Poland. It is upon the US government to save the honor of Western democracies as promoters of peace, legality and moderation. Return to the February 21 Accords before the hell of war breaks loose!
Editors notes: International Action Center founder Ramsey Clark, a former US Attorney General and internationally renown human rights defender, received the respected United Nations Prize in the Field of Human Rights on the 60th Anniversary of the Universal Declaration of Human Rights at United Nations Headquarters in New York on 10 December 2008.
Article published with permission from Ramsey Clark.


Ella does some field research at the New Hampshire State House.
Ella Lubell has me on the edge of my seat, and she's telling a story that's 150 years old.
Ella is a Norwich resident and Hanover High graduate who is leaving town in a few days to begin her studies at Yale. She spent a chunk of the summer in the company of a fellow town resident, Albert B. Nye, gently tracing her finger over lines he wrote in wartime circumstances difficult to imagine today.
Nye was a Civil War soldier, husband and father. He left a diary of his experiences, now in the collection of the Norwich Historical Society -- "a small, leather bound, unassuming journal in an acid free box," as Ella put it.
Her mission: transcribing Nye's diary. She loved it. "It's very meditative," she said. "It feels a little bit like a conversation."
The result: a series of blog posts Ella wrote that will walk readers through Nye's account, unfolding the story of his wartime service as he lived it.
I'm passionate about history myself, so talking with Ella about her project was a pleasure. Her blog is too. (You can access it here.) It is interactive, rich with context and full of insight. And she spins a pretty good yarn.
As of this writing, she has Nye in the New York Hospital in the winter of 1862, struggling to overcome a case of jaundice -- "an entire continent away from the rest of his regiment, waiting to be released and scouring the city for anyone who can help him get to New Orleans."
What awaits him is a struggle with an unfamiliar climate, death and disease -- and battle.
Ella's remaining posts -- she is planning 10, and had eight written when we spoke yesterday -- will be published on Saturdays and Sundays through the rest of August. By the time the last of the posts appears, she'll be beginning a first year of studies that will immerse her in literature, philosophy, history and political thought.
While Ella is away, work on transcribing the diary -- she didn't quite finish -- will continue, and in time that effort will become a story in itself. The historical society's hope is to crowd-source the work: to post images of the remaining pages online for people to claim and transcribe themselves.
I asked her if she could actually have a conversation with Albert Nye, what she'd ask.
"I would want to know how the war affected his life after," she said. He returned home -- as soldiers do today -- to a wife and child, and resumed civilian life. Eventually he became a house painter. "I would just want to ask how those different versions of himself fit together."
At least one of those "versions of himself" was, of course, extraordinary. Now, through Ella's efforts, and in time the community's, all of us can share in it.


text:
HP PC Send Fax Driver
Use this driver to conveniently fax documents from within PC applications through an analog-fax enabled HP multi-function (MFP) device.
Version: 61.063.666.41
Relase Date : 10 Jun 2013
File name: HP_MFP_SendFax300_32bit.exe
Microsoft Windows XP (32-bit)
Microsoft Windows Vista (32-bit)
Microsoft Windows Server 2008 W32
Microsoft Windows 7 (32-bit)
Microsoft Windows Server 2008 R2
Microsoft Windows 8.1 (32-bit)
File Size : 28.03Mb
Hp Critical Update. Cannot find scanned image
Hp Update
Critical Update: Cannot find scanned image or document files after scanning completes.
When scanning after installing or upgrading to HP Solution Center 13, scanning appears to complete correctly, however scanned images or documents cannot be found on the computer.
File name: CPE_SCAN_DESTINATION_UPDATE_hpcom_001_003.exe
Fix and enhancements
Corrects a scan file configuration problem which causes scan files to be sent to a nonexistent folder.
Universal PCL5 Print Driver for HP v5.9.0.18326
HP Universal Print Driver for Windows PCL6 (64/32-bit)
This is the most current PCL6 driver of the HP Universal Print Driver (UPD) for Windows 32 bit and 64 bit systems.
The HP Universal Print Driver (HP UPD) is a single driver that gives users access to a range of HP print devices in the office or on the road without downloading separate drivers for every utilized printer.
Available for both 32-bit and 64-bit operating systems
HP Universal Print Driver v7.0.1.24923 Release
The HP Universal Print Driver (HP UPD) is a single driver that gives users access to a range of HP print devices in the office or on the road without downloading separate drivers for every printer in use.
The HP UPD works well with a broad range of networked and direct-connected HP print products using PCL 6 or PostScript emulation. The HP UPD installs in Traditional Mode, or Dynamic Mode to enhance mobile printing. Dynamic Mode discovers network printers or enter a printer name or IP address and print anywhere on your network. It is great when you are on the road or want to print to a new device but do not have a model specific print driver installed. Traditional Mode is just like a traditional driver. Both modes can be used on the same PC.
Release details HP LaserJet
Relase Date : Oct 6, 2021
Microsoft Windows 10 (32-bit), Microsoft Windows 7 (32-bit), Microsoft Windows 8 (32-bit), Microsoft Windows 8.1 (32-bit), Microsoft Windows Server 2012, Microsoft Windows Server 2016, Microsoft Windows Server 2019
File name: upd-pcl6-x32-7.0.1.24923.exe
HP Universal Print Driver for Windows PostScript (64-Bit / 32-Bit)
This is the most current PostScript emulation driver of the HP Universal Print Driver (UPD) for Windows 32/64 bit systems; Level 2 and 3 emulation.
Operating Systems: Microsoft Windows 7 (32-bit), Microsoft Windows XP, Microsoft Windows Vista (32-bit), Microsoft Windows 8 (32-bit), Microsoft Windows 8.1 (32-bit), Microsoft Windows 10 (32-bit)
File name: upd-ps-x32-7.0.1.24923.exe
File Size : 19,62 Mb
Operating Systems: Microsoft Windows 8 (64-bit) , Microsoft Windows 7 (64-bit), Microsoft Windows 8.1 (64-bit), Microsoft Windows Server 2019, Microsoft Windows 10 (64-bit)


text:
Little festive joy for single parents this Christmas
Equifax reveals a third of single parents feel less confident about meeting their credit commitments over the Christmas period.
According to research conducted by Equifax UK, nearly 60% of single parents felt less confident about their finances in the run-up to Christmas, compared to other times during the year.
After Christmas 2018 and coming into 2019, nearly over 31% of single parents polled felt less confident about meeting their credit commitments and only 23% felt more confident about their financial situation and managing bills; it is clear to see that the cost of Christmas remains to be a financial burden for parents year on year.
Although Christmas can be a wonderful time of year filled with celebration it can also be very stressful for parents. Single parents, in particular, can feel the pressure of managing their children's expectations.
According to research conducted by Equifax UK, 63% of single parents polled felt under pressure to give their children everything they want at Christmas – even if it meant damaging their finances.
With only 8% of parents feeling they had their money situation under control and 21% of parents admitting to not knowing how they would finance Christmas at all, Equifax has devised an educational infographic which offers practical tips to help single parents with their finances over the festive period.
Searches across the internet around the terms 'Christmas on a budget' and 'Christmas gifts on a budget' appear to increase significantly in the months leading up to Christmas.
Over November and December last year, more than 10,000 monthly average searches were made around the term 'Christmas on a budget'.
Lisa Hardstaff consumer credit information expert at Equifax, said "All of the excitement in the run-up to Christmas can sometimes be overshadowed for many single parent families who are struggling to cope with all the seasonal spending.
"Indeed a lot of people may have to rely on credit cards to ease the financial strain, meaning they are likely to go into the New Year with debts.
"This can be even more pronounced for single parents, who only have one source of income.
"Whilst sometimes easier said than done, budgeting and planning for big expenses plays a vital role to help ease the financial burden.
"The crucial element when it comes to budgeting is to gather as much information about your financial situation as possible to see where you might be able to cut back on some areas.
"Even small things like buying a morning coffee on the way to work can all add up!
"Taking small steps to save where possible could really help to ease the financial strain during the festive period this year."
SourceEquifax


The Awesoming, now on DVD! Featuring 4 New Nostalgia Critic reviews previously only available through Vimeo! As well as a bunch of bonus features, 2 new commentaries and more! This DVD is region free and will work on any DVD player! Shipping begins immediately.


Leukeran belongs to the group of medications known as antineoplastics. It contains chlorambucil. This medication works by preventing the growth of cancer cells by interfering with the genetic material DNA, which is necessary for reproduction of cells. Leukeran is used to treat certain types of cancer (leukemia, lymphoma, Hodgkin's disease). This drug may also be used for certain blood diseases, certain types of kidney disease in children that have not responded to other treatment.
Take Leukeran exactly as prescribed by your doctor. Take this medication by mouth. Drink plenty of fluids while taking this medication to prevent side effects. For chronic lymphocytic leukemia, the usual adult starting dose is 0.15 mg per kilogram body weight per day until the total white blood cell count reaches the desired range. 4 weeks after the first course of treatment, the medication may be restarted at a dose of 0.1 mg per kilogram body weight per day. For Hodgkin's disease, the usual dose is 0.2 mg per kilogram body weight per day for 4 to 8 weeks. For non-Hodgkin's lymphoma, the usual dose is 0.1 mg to 0.2 mg per kilogram body weight per day for 4 to 8 weeks initially.
Before taking Leukeran you should talk with your doctor if you have seizures, head injury, gout, kidney stones, blood disorders (anemia, neutropenia, thrombocytopenia). Avoid contact with people who have contagious infections. Do not have immunizations or vaccinations without the consent of your doctor.
Do not take Leukeran if you are allergic to chlorambucil or any ingredients of this medication, are pregnant or breastfeeding, or if you have had a full course of radiation or chemotherapy within the previous 4 weeks.
Get emergency medical help if you have changes in menstrual period, diarrhea, nausea, vomiting, muscle twitching, seizures, skin rash, tremors, unexpected bleeding, difficult urination, unusual tiredness, yellow eyes or skin, black or bloody stools, hives, difficulty breathing, swelling of the throat, mouth, joint pain, pale skin, stomach pain, loss of appetite, dark urine, sore throat, weakness, severe dizziness. If you notice other effects not listed above, contact your doctor.
Tell your doctor about all other medications you use, especially: live vaccines (mumps, measles, yellow fever), natalizumab, phenylbutazone, pimecrolimus, roflumilast. Interaction between two medications does not always mean that you must stop taking one of them. Tell your doctor or prescriber about all prescription, over-the-counter, and herbal medications you are taking.
If you think you have overdosed the medicine seek emergency medical help at once. The overdose symptoms are loss of balance, excitation, convulsions.


This entry was posted on May 9, 2017, 1:29 am and is filed under education, english, personal, reflections, school, university. You can follow any responses to this entry through RSS 2.0. You can leave a response, or trackback from your own site.