text:
\section{Introduction}

We investigate the dynamics of a relativistic charged particle with charge $q \neq
0$ that moves on $\bfR^n$, $n \in \bfN$, and is influenced by homogeneous time-dependent electric fields $\bfE(t) = E(t) = (E_1(t), ... , E_n(t)
)$, which satisfy $E_j(t) \in C^{1} (\bfR )$ for all $j \in
\{ 1,...,n\}$ and
\begin{align} \label{E1}
 \sup _{t \in \bfR}
\sum_{j=0}^n|E_j ^{(k)}(t)| < E_{0,k}, \quad E_j^{(k)} (t) =
\frac{d^k E_j (t)}{dt^k}, \quad k \in  \{0,1\}, \quad j \in \{ 1,...,n\},
\end{align}
 where $0< E_{0,k} < \infty $ is a constant. The wave functions under
 consideration satisfy the following Klein-Gordon
equations:
\begin{align}\label{1}
&\begin{cases}
&(i \partial_t + q_E )^2 \psi_0 (t,x) = L(0,p) \psi_0 (t,x), \\
& \psi_0(0,x) = \psi_{0,0}, \quad \left\{
(i \partial _t + q_E )(\psi_0(t,x))
\right\} |_{t=0} = \psi_{0,1},
\end{cases}  \\ & L(0,p) = c^2p^2 +
 (mc^2)^2, \quad q_E=q_E(t,x) = qE(t) \cdot x,
\nn
\end{align}
where $x =(x_1,...,x_n)\in \bfR
^n$, $p =(p_1,...,p_n) = -i (\partial{x_1},....,\partial{x_n})  = -i \nabla$, $m >0$, and $q \in \bfR
\backslash \{0\}$ are the
position, momentum, mass, and charge of the charged particle, respectively. We let
$c>0$ denote the speed of light; the inner product of $a,\, b \in \bfR
^n$ is denoted by $a \cdot b$.
To introduce
the main theorem, we consider the system of Veseli\'{c} \cite{Ve}.

Let $\psi_0(t,x)$, $\psi _{0,0}$, and $\psi_{0,1}$ be equivalent to those
 in \eqref{1}. The substitutions $\psi_{0,1} (t,x) = (i \partial _t +q_E)
 \psi_{0} (t,x)$ and
\begin{align*}
\Psi_0(t,x) := \MAT{\psi_0(t,x) \\ \psi_{0,1} (t,x)} , \quad \Psi_{0} =
 \Psi_{0} (0,x) = \MAT{\psi_{0,0 } \\ \psi_{0,1}}
\end{align*}
yield the following (Hamilton) system:
\begin{align} \label{M4}
i \frac{\partial}{\partial t} \Psi _0 (t,x) = A_0(t) \Psi_0(t,x), \quad
A_0(t) := \MAT{-q_E & 1 \\ L(0,p) & -q_E}, \quad \Psi_0(0,x) = \Psi_0.
\end{align}
By substituting $\SCR{K}_{\alpha}$, defined in \eqref{Q0} (see also \cite{Ve},
 (1.3)), and by using the same scheme
as that found in \cite{Ve}, we arrive at the following system on
$\SCR{H} = L^2(\bfR ^n) \times L^2(\bfR ^n)$:
\begin{align}\label{Q1}
i \frac{\partial}{\partial t} \Phi_{0, \alpha}(t,x) = H_{0, \alpha}(t)
 \Phi_{0, \alpha} (t,x), \quad
\Phi_{0,\alpha}(0,x)= \Phi_{0, \alpha} \in \SCR{H},
\end{align}
where $H_{0,\alpha}(t) = \hat{H}_0(t) + P_{0, \alpha} (t)$ with
\begin{align} \label{Q6}
\hat{H}_{0}(t) &= \MAT{-q_E & (L(0,p))^{1/2} \\ (L(0,p))^{1/2} & -q_E} ,
 \\
P_{0, \alpha}(t) &= \frac{ic^2}{2} \MAT{(1-2 \alpha)qE(t) \cdot p (L(0,p))^{-1} & 0
 \\ 0 & - (1+2 \alpha)qE(t) \cdot p (L(0,p))^{-1}}, \label{Q7}
\end{align}
where $i = \sqrt{-1}$ and
\begin{align*}
\Phi_{0, \alpha}(t,x) = \MAT{(L(0,p))^{1/4- \alpha /2} \psi_0(t,x) \\
 (L(0,p))^{-1/4 - \alpha /2} \psi_{0,1} (t,x)}.
\end{align*}
 The construction scheme of $H_{0, \alpha} (t)$ can be found in Appendix
 A or \cite{Ve}. Here, we call $U_{0, \alpha} (t)$ the propagator for $H_{0, \alpha}
(t)$ if $U_{0, \alpha} (t)$ satisfies the following equations:
\begin{align*}
& i \partial _t U_{0, \alpha} (t) = H_{0,\alpha}(t) U_{0, \alpha}(t) ,\quad  i \partial _t
 (U_{0, \alpha} (t))^{-1} = -(U_{0, \alpha}(t))^{-1} H_{0,\alpha}(t), \\
& U_{0, \alpha} (t) (U_{0, \alpha} (t))^{-1} = U_{0, \alpha} (0) = \mathrm{Id}_{\SCR{H}}.
\end{align*}
The solution of
\eqref{Q1} is denoted by $\Phi_{0, \alpha} (t,x) = U_{0, \alpha}(t) \Phi_{0, \alpha}$.
The main theorem of this paper proves that $ 0< \|U_{0, 0} (t)
\|_{\SCR{B}(\SCR{H})} < \infty$ as $t \to \infty$ and that for any $\alpha \neq 0$ and $\varphi \in C_0^{\infty} ({\bf R}^n)$,
$\left\| U_{0, \alpha} (t)  \varphi (p)\right\|_{\SCR{B}(\SCR{H})} \to 0$ or $\infty$
as $t \to \infty $, where $\SCR{B} (\SCR{H})$ is the operator
norm on $\SCR{H}$. First, we analyze the asymptotic behavior of
$U_{0, \alpha}(t)$ in $t$. Unfortunately, $U_{0, \alpha} (t)$ is difficult to control for general electric fields satisfying only \eqref{E1}. Hence,
 we impose the following additional condition {\bf (E1)} on electric fields: \\ \\
{\bf (E1):} Let $E(t)$ satisfy \eqref{E1}, and define $b(t) = \int_0^t
qE(s) ds$. Then $b(t)$ satisfies $\lim_{t \to \infty} |b(t)| \to \infty$.
Moreover, for any vector $a \in {\bf R}^n$, there exist constants $e_0$
and $e_1$, independent of $t$ and $a$, such that
\begin{align}\label{14}
\int_{|a+ b(s)| \leq 2E_{0,0}/(mc^2)} |b'(s)|   ds
  \leq e_0, \quad
\int_{0}^t \frac{|b'(s) |^2 + |b''(s)|}{c^2 (a + b(s))^2 + (mc^2)^2} ds
 \leq e_1
\end{align}
holds.

Models of electric fields satisfying Assumption {\bf (E1)} and
remarks regarding this assumption can be found in Appendix B.

We define the Fourier transform $\SCR{F}_1^{+1}$ and inverse Fourier transform
$\SCR{F}_1^{-1} $ on $L^2(\bfR ^n)$ as follows:
\begin{align} \label{13}
\begin{array}{lll}
\hat{\phi}(q) &:= \SCR{F}_1^{+1}[\phi](q), \\
\check{\phi}(q) &:= \SCR{F}_1^{-1} [\phi](q),
\end{array}
\quad
\SCR{F}_1^{\pm 1} [\phi] (q) :=
(2 \pi i)^{-n/2} \int_{\bfR^n} e^{\mp i q \cdot \eta} \phi(\eta) d \eta.
\end{align}
We now state the main theorem in this paper.
\begin{theorem} \label{T2}
Set $\alpha =0$ in \eqref{Q1} and \eqref{Q7}, and suppose Assumption {\bf
 (E1)} holds. Then for all $t \in {\bf R}$, there exist $0< \Gamma _1 < \Gamma _2$, independent of $t$, such that
\begin{align}\label{41}
\Gamma _1 \leq \left\|
U_{0,0}(t)
\right\|_{{\mathscr B}({\mathscr H})} \leq \Gamma _2
\end{align}
holds, where ${\mathscr B} (\mathscr H)$ is the operator norm on
 $\SCR{H}$. Conversely, for any $\alpha \neq 0$ and $\Phi_{0, \alpha}
 \in \SCR{F}_1^{-1}C_0 ({\bf R}^n) \times \SCR{F}_1^{-1}C_0({\bf R}^n)$,
\begin{align*}
\lim_{t \to \infty} \left\| U_{0, \alpha}(t) \Phi_{0, \alpha}
 \right\|_{\SCR{H}} =
\begin{cases}
 0,  & \alpha >0, \\
\infty, & \alpha <0,
\end{cases}
\end{align*}
holds.
\end{theorem}
Herein, we say $U_{0,0}(t)$ is stable on $\SCR{H}$ and $U_{0, \alpha}(t)$,
$\alpha \neq 0$ is unstable on $\SCR{H}$. As a corollary to Theorem \ref{T2}, we obtain the
following inequality.

\begin{corollary}\label{Q8}
Suppose Assumption {\bf (E1)} holds. Then for all $t \in {\bf R}$, ${\psi}_{0,0} \in
 H^{1/2} ({\bf R}^n)$, and ${\psi}_{0,1}
 \in H^{-1/2} ({\bf R}^n)$, there exist $0< \gamma _1< \gamma _2$ independent
 of $t$ such that
\begin{align*} \nn
& \gamma _1
\left(
 \left\| (L(0,p))^{1/4} \psi_{0,0} \right\|_{L^2(\bfR ^n)}^2 +
 \left\|
(L(0,p))^{-1/4} \psi_{0,1}
\right\|_{L^2(\bfR ^n)}^2
\right)
\\  & \qquad < \left\| (L(0,p))^{1/4} \psi_0 (t,x)\right\|_{L^2(\bfR ^n)}^2 +
 \left\|
(L(0,p))^{-1/4} (i \partial _t + q_E)\psi_0(t,x)
\right\|_{L^2(\bfR ^n)}^2 \\ & \qquad \qquad
< \gamma _2 \left(
 \left\| (L(0,p))^{1/4} \psi_{0,0}\right\|_{L^2(\bfR ^n)}^2 +
 \left\|
(L(0,p))^{-1/4} \psi_{0,1}
\right\|_{L^2(\bfR ^n)}^2
\right) \nn
\end{align*}
holds, where $\psi_0(t,x)$, $\psi_{0,0}$, and $\psi_{0,1}$ are the same
 as those in \eqref{1}.
\end{corollary}

If $q_E$ is independent of time and satisfies $$ \left\| q_E u
 \right\|_{L^2(\bfR ^n)} \leq  \delta \left\|   (L(0,p))^{1/2} u
 \right\| _{L^2(\bfR ^n)}, \quad 0< \delta <1, $$ then
 Najman \cite{Na} showed that $A_0 (t)$ generates a uniformly bounded
 propagator on $\SCR{K}_{\alpha}$. Veseli\'{c} subsequently applied Najman's scheme
 to non-decreasing constant electric fields $qE \cdot x$ and obtained
 stability in the time-evolution operator on $\SCR{K}_{0}$ and
 instability on $\SCR{K}_{\alpha}$, where $\alpha \neq 0$. Essential to the proof is the factorization of propagator $U_{0,
 \alpha }(t) = V_{\alpha} e^{-itqE \cdot x} V_{\alpha} ^{-1} $, where
 $V_{\alpha}$ is a time-independent linear operator satisfying
 differential equations (see \cite{Ve}). By virtue of this factorization,  $\left\|U_{0, \alpha} (t) \Phi \right\|$ can be estimated by analyzing $V_{\alpha}$
 instead of $e^{-itH_{0, \alpha}}$. We try to extend this approach to
 time-dependent electric fields. First, we form
 another factorization of $U_{0, \alpha} (t)$ since the aforementioned $V_{\alpha}$ depends on time if the electric fields depend on
 time (i.e., $V_{\alpha}(t) e^{ib(t) \cdot x} V_{\alpha}
 (t)^{-1}$ is different from $U_{0, \alpha} (t)$). In order to form a
 new factorization, we focus on the so-called {\em Avron-Herbst formula}. We refer to Avron-Herbst \cite{AH} and Cycon-Froese-Kirsch-Simon
 \cite{CFKS}, Theorem 7.1., which consider the study of the Schr\"{o}dinger equations with time-dependent (and constant) electric fields:
\begin{align}\label{st}
\begin{cases}
i \partial_t u(t,x) &= ( p^2 /(2m) -qE(t) \cdot x) u(t,x) =:
 H_0^{\mathrm S} (t) u(t,x) ,\\
u(0,x) &= u_0,
\end{cases}
\end{align}
where $H_0^{\mathrm S} (t)$ is the {\em Stark Hamiltonian}. For a solution $u(t,x)$ to \eqref{st}, substituting $u(t,x) = e^{ib(t) \cdot x} v(t,x) $ yields $ i \partial _t v(t,x) = (p+ b(t))^2v(t,x)/(2m)$. Thus, by letting
$$
v(t,x) = e^{-ia(t)}e^{-ic(t) \cdot p}e^{-itp^2/(2m)}u_0 , \quad a(t) = \int_0^t \frac{b(s)^2}{2m} ds , \quad c(t) = \int _0^t \frac{b(s)}{m} ds,
$$
one obtains $u(t,x) = e^{-ia(t)} e^{ib(t) \cdot x} e^{-ic(t) \cdot p}
 e^{-it p^2/(2m)}u_0$, i.e., a propagator for $H_0^S(t)$ can be
 described by $ e^{-ia(t)} e^{ib(t) \cdot x} e^{-ic(t) \cdot p} e^{-it
 p^2/(2m)}$. This factorization of the propagator is called the
 Avron-Herbst formula. This factorization has been applied to many research areas such as quantum scattering
theory and non-linear analysis (see Adachi-Ishida
 \cite{AI}, Avron-Herbst \cite{AH}, M\o ller \cite{Mo}, and
 Carles-Nakamura \cite{CN}). We attempt to apply this scheme to
 \eqref{1}; in this process, we analyze the differential equation $-\partial_t^2 u_0 (t,x) = ({c^2(p+b(t))^2 + (mc^2)^2}) u_0 (t,x)$.
 To consider the asymptotic behavior of solutions to this equation, we
 use the approach of Hochstadt \cite{Ho1}. At the conclusion of this paper
 (\S{4.1} \eqref{AV}), we obtain a new factorization of the
 propagator $U_{0,\alpha} (t)$.

Our first approach to prove Theorem \ref{T2} is to reduce
\eqref{1} to the ordinary differential equation in \eqref{5} through
the Fourier transform. A
similar approach to the case where the potential $q_E$ is
 dependent on time but independent of $x$, was studied by B\"{o}hme-Ressig \cite{BR1}, \cite{BR2}. Time-decaying dissipative wave equations were
studied by Wirth \cite{W1}, \cite{W2}. Our approach may be applicable
to such equations and other open problems such as those discussed by
Todorova-Yordanov \cite{TY}.

\section{Definitions and notation}

In this section, we introduce definitions and notation. Let $C$ be a constant where $C>0$. For $h(\tau, \eta)$,
$\tau \in \bfR$, and $\eta \in \bfR ^n$, let $h'(\tau, \eta)$ and $h''(\tau, \eta)$ be defined by
$$
h'(\tau , \eta) := h^{(1)} (\tau, \eta), \quad h''(\tau, \eta) :=
h^{(2)} (\tau , \eta), \quad h^{(l)} (\tau , \eta) := \frac{\partial^l
h}{\partial \tau ^l} (\tau, \eta),
$$
for $l \in \{0,1,2\}$. Moreover, let $\SCR{H}=L^2(\bfR
^n) \times L^2(\bfR ^n)$.  For $ \Phi =  (\phi_1 ,
\phi _2)^{\mathrm{T}} $ and $\Psi = {(\psi _1,  \psi _2 )^{\mathrm{T}}}$, the norm of the Hilbert space $\SCR{H}$ is defined by
 $\left\| \Phi
\right\|_{\SCR{H}}^2 = \left\| \phi_1 \right\|_{L^2(\bfR ^n)}^2 + \left\|
\phi _2
\right\|_{L^2(\bfR ^n)}^2$ and inner product of $\SCR{H}$ is defined by
\begin{align*}
\left( \Phi , \Psi \right)_{\SCR{H}} :
= (\phi_1, \psi _1)_{L^2(\bfR ^n)}
 + ( \phi_2, \psi _2)_{L^2(\bfR ^n)}.
\end{align*}
Let $A$, $B$, $C$, and $D$ be linear
 operators on $L^2(\bfR ^n)$, and let
\begin{align*}
\CAL{A} = \MAT{A & B \\ C & D}.
\end{align*}
Then for $\Phi = (\phi_1,
  \phi_2)^{\mathrm{T}}$, we define
\begin{align*}
\CAL{A} \Phi = \MAT{A & B \\ C  & D} \MAT{\phi _1 \\ \phi_2 } := \MAT{A \phi_1 +B \phi_2
 \\ C \phi_1 + D \phi_2},
\end{align*}
so that $\CAL{A}$ is a linear operator on $\SCR{H}$. Furthermore, for $\Phi _1 \in \SCR{H}$ and $\Phi _2 \in \SCR{H}$, if
there exists  $\Phi_3 \in \SCR{H}$ such that
\begin{align*}
(\CAL{A} \Phi_1 , \Phi_2)_{\SCR{H}} = (\Phi_1, \Phi_3)_{\SCR{H}}
\end{align*}
holds, then we define $\Phi _3 = \CAL{A}^{\ast} \Phi_2$; it can be
easily obtained by
\begin{align*}
\CAL{A} ^{\ast} = \MAT{A & B \\ C & D} ^{\ast} = \MAT{A^{\ast} & C^{\ast} \\ B^{\ast} &
 D^{\ast}},
\end{align*}
where $A^{\ast}$, $B^{\ast}$, $C^{\ast}$, and $D^{\ast}$ are the
adjoint operators of $A$, $B$, $C$, and $D$, respectively, on $L^2(\bfR
^n)$.
Finally, $(A)_M \Phi$ for $\Phi \in
\SCR{H}$ means
\begin{align} \label{X10}
 (A)_M \Phi = \MAT{A & 0 \\ 0 & A} \Phi
\end{align}
 for some linear operator $A$ on $L^2(\bfR ^n)$.

\section{Estimates of solutions to \eqref{1} }

First, we define $b(t) = (b_1
(t), b_2(t), ...,b_n(t))$ as
$\int_0^t qE(s) ds$ and take
$$ u_0 (t,x) = e^{-i \int_0^t qE(s) \cdot x ds
} \psi_0(t,x)= e^{ -ib(t) \cdot x} \psi _0(t,x). $$ Then $u_0
(t,x)$ satisfies equations
\begin{align} \label{L}
& \begin{cases}
& - \partial _t ^2 u_0 (t,x) = L(t,p) u_0 (t,x), \\
& u_0 (0,x) = \psi_{0,0}, \quad (i \partial _t u_0 )(0,x) = \psi
 _{0,1},
\end{cases}
\\ \nn
& L(t,p) := c^2 (p + b(t))^2 + (mc^2)^2,
\end{align}
where $e^{-ib(t) \cdot  x} (i \partial _t) e^{ib(t) \cdot x} =
i \partial _t - qE(t) \cdot x$ and $e^{-ib(t) \cdot  x} p e^{ib(t) \cdot x} =
p +b(t)$ hold on the test function.
By the Fourier transform in \eqref{13}, \eqref{L} is transformed into
\begin{align} \label{z1}
(\partial _t ^2 \hat{u}_0)(t,\xi) + L(t, \xi)\hat{u}_0 (t, \xi) = 0,
 \quad
\hat{u}_0(0,\xi) = \hat{\psi}_{0,0}, \quad
(i\partial _t \hat{u}_0) (0, \xi) = \hat{\psi}_{0,1}.
\end{align}
Hence, for any fixed $\xi \in {\bf R}^n$, define $\zeta _j (t, \xi)$, $j \in \{0,1\}$, as the solution to
\begin{align}
\zeta _j ''(t, \xi) + L(t,\xi) \zeta _j (t, \xi) = 0, \quad
\begin{cases}
\zeta _0 (0, \xi)=1, \\
\zeta' _0 (0, \xi)= 0,
\end{cases}
 \quad
\begin{cases}
\zeta _1 (0, \xi)=0, \\
\zeta' _1 (0, \xi)= 1.
\end{cases}
\label{5}
\end{align}
Note that the solutions of \eqref{1} can be written as
\begin{align} \label{L4}
\psi _0 (t,x) = e^{ib(t) \cdot x} \SCR{F}^{-1}_1 \zeta _0 (t, \xi)
 \hat{\psi}_{0,0} + e^{ib(t)\cdot x} \SCR{F}^{-1}_1 \zeta _1 (t, \xi)
 \hat{\psi}_{0,1}.
\end{align}

\subsection{Hochstadt type solutions}

Let $\psi_0 (t,x)$ be
a solution to the Klein-Gordon equations in \eqref{1}. Noting \eqref{L4}, it
is equivalent to analyze the asymptotic behavior of the solution to
\eqref{5} and analyze the asymptotic behavior of the solution to \eqref{1}. To analyze \eqref{5}, we consider the approach of
Hochstadt \cite{Ho1} (also, see Hochstadt \cite{Ho2}). For simplicity, we
denote
\begin{align}
L(t,\xi)^{1/2} = Q(t,\xi), \quad L(t,p)^{1/2} = Q(t,p) \label{Q20}
\end{align}
in the following. Suppose that $\zeta _j
(t, \xi)$ and $\zeta _j '(t, \xi)$ are represented by
\begin{align}   \label{L5}
& \begin{cases}
&\zeta _0 (t,\xi) = A(t,\xi) \COS{B(t,\xi)} , \ \zeta_0' (t,\xi) =
 -A(t,\xi)Q(t,\xi) \SIN{B(t, \xi)},  \\
\ &A(0, \xi)=1 , \ B(0, \xi)=0,
\end{cases}
\\
& \begin{cases}
&\zeta _1 (t, \xi) = C(t,\xi) \SIN{D(t, \xi)} , \ \zeta_1' (t, \xi)
 = C(t, \xi)Q(t, \xi) \COS{D(t, \xi)}
, \\ &C(0, \xi) = Q(0, \xi)^{-1} , \ D(0, \xi)=0,
\end{cases}
\label{L6}
\end{align}
respectively, for functions $A(t, \xi)$, $B(t, \xi)$, $C(t, \xi)$,
and $D(t, \xi)$. Considering \eqref{5}, \eqref{L5}, and \eqref{L6}, we obtain differential equations
\begin{align} \label{6}
&\begin{cases}
&A'(t,\xi) = - Q(t,\xi)^{-1}Q'(t,\xi) \sin ^2 ({B(t,\xi)}) A(t,\xi), \\
&B'(t,\xi) = Q(t,\xi) -  Q(t,\xi)^{-1}Q'(t,\xi) \SIN{B(t,\xi)}\COS{B(t,\xi)},
\end{cases}
\\ \label{7}
&\begin{cases}
&C'(t,\xi) = - Q(t,\xi)^{-1}Q'(t,\xi) \cos ^2 ({D(t,\xi)} ) C(t,\xi), \\
&D'(t,\xi) = Q(t,\xi) +  Q(t,\xi)^{-1}Q'(t,\xi) \SIN{D(t,\xi)}\COS{D(t,\xi)}.
\end{cases}
\end{align}
Equations \eqref{6} and \eqref{7} yield
\begin{align}\nonumber
A(t,\xi) &= e^{- \int_0^t Q(s,\xi)^{-1}Q'(s,\xi) \sin ^2 ({B(s,\xi)}) ds  }, \\
C(t,\xi) &= Q(0 , \xi)^{-1} e^{- \int_0^t Q(s,\xi)^{-1}Q'(s,\xi)
 \cos ^2 ({D(s,\xi)})  ds},  \label{8}
\end{align}
and
\begin{align} \nonumber
B(t,\xi) &= \int_0^t Q(s,\xi) -  Q(s,\xi)^{-1}Q'(s,\xi) \SIN{B(s,\xi)}\COS{B(s,\xi)} ds, \\
D(t,\xi) &= \int_0^t Q(s,\xi) +  Q(s, \xi)^{-1}Q'(s,\xi)
 \SIN{D(s,\xi)}\COS{D(s,\xi)} ds.  \label{K3}
\end{align}

\begin{lemma}
Functions $B(t) = B(t, \cdot)$ and $D(t) = D(t, \cdot)$ ($A(t, \cdot)$ and $C(t,
 \cdot)$) are in $C^{2} ({\bf R})$. Moreover, $B(t)$ and $D(t)$
 satisfying the integral equation \eqref{K3} are unique.
\end{lemma}
\begin{proof}
It is obvious that $B(t)$ and $D(t)$ are included in $C^{2}({\bf R})$ since $b(t) \in C^2 ({\bf R})$ (i.e., $Q(t, \cdot)$ and $Q'(t,
 \cdot)/Q(t, \cdot) $ are in
 $C^1({\bf R})$). Hence, we only
 prove the uniqueness of $B(t)$ and $D(t)$. Further, we only prove the uniqueness of
 $B(t)$ since the uniqueness of $D(t)$ can be proven in the same way.

First, we prove that for all $t$ and $\xi$, if $B_1(t,\xi)$ and $B_2(t, \xi)$
 satisfy \eqref{K3}, then $B_1 (t, \xi) = B_2 (t, \xi)$. Let $\ep_E
 < mc/ (E_{0,0})$ and $0\leq t \leq \ep_E$. Then by \eqref{K3} and
\begin{align}\label{t2}
\left|
\frac{Q'(s, \xi)}{Q(s, \xi)}
\right| = \left|
\frac{c^2(\xi+b(s)) \cdot b'(s)}{c^2 (\xi+b(s))^2 + (mc^2)^2}
\right| \leq \frac{c |b'(s)|}{Q(s, \xi)} \leq \frac{c|b'(s)|}{mc^2},
\end{align}
we have
\begin{align}
\left| \nn
B_1(t, \xi) - B_2 (t, \xi)
\right| &=
\left|
\int_0^t \frac{Q'(s, \xi)}{2Q(s, \xi)} \left(
\int_{B_2(s, \xi)}^{B_1 (s, \xi)} \frac{d}{d \tau} \sin (2 \tau) d \tau
\right) ds
\right| \\  & \leq \frac{E_{0,0}}{ mc} \int_0^t \left|
\int_{B_2(s, \xi)}^{B_1 (s, \xi)} \cos (2\tau) d \tau
\right| ds.  \label{tt6}
\end{align}
With \eqref{tt6} and since $ t \leq \ep _E <mc/E_{0,0} $, it follows that $\sup_{t \leq
 \ep_{E}, \xi \in {\bf R}} |B_1 (t, \xi) - B_2 (t, \xi)|=0$. For $t \leq
 2 \ep_E$, we have
\begin{align*}
& \sup_{\ep_E \leq t \leq 2 \ep_E, \ \xi\in {\bf R}}|B_1(t, \xi) - B_2
 (t, \xi)| \\ & \quad \leq
 \frac{E_{0,0}}{mc} \left(
 0 + \sup_{\ep_{E} \leq t \leq 2 \ep_{E}, \ \xi\in {\bf R}}
 \int_{\ep_E}^t |B_1(s, \xi) - B_2 (s, \xi)| ds
\right) .
\end{align*}
This also implies $\sup_{\ep_E \leq t \leq 2\ep_E, \xi} |B_1(t,
 \xi)-B_2 (t, \xi)| =0$. By repeating the same calculation for $t \in [n
 \ep _E, (n+1)\ep_E]$ with $n \in {\bf N}$, the lemma holds.
\end{proof}
First, we impose
$\hat{\psi}_{0,j}  \in C_0(\bfR ^n)$, $j \in \{0,1\}$, in \eqref{1} and define $\varphi_{j} \in C_0^{\infty}( \bfR ^n)$ such that
\begin{align}\label{K7}
\varphi_0 (\xi) =
1, \  \mbox{on the support of $\hat{\psi}_{0,0} (\xi)$} , \quad
\varphi_1 (\xi) =
1, \   \mbox{on the support of $\hat{\psi}_{0,1} (\xi)$}.
\end{align}
Noting \eqref{5}, \eqref{L5}, \eqref{L6},
\eqref{8}, $|Q(t_0, \xi)|^{-1} \leq C$, $|Q(0, \xi)|^{-1} \leq C$,
and the fact that $|Q(t_0, \xi)| \leq C$ holds on the support of $\varphi_j
(\xi)$, the following proposition immediately holds.
\begin{proposition} \label{Q3}
Let $\zeta _0 (t, \xi)$ and $\zeta _1 (t, \xi)$ be equal to those defined in
 \eqref{L5} and \eqref{L6}, respectively, and let $\varphi _0 (\xi)$ and
 $\varphi_1 (\xi)$ be equal to those defined in \eqref{K7}. Then for every fixed $t_0
 \in \bfR$,
\begin{align}
\sup_{\xi \in {\bf R}^n} |\zeta _j ^{(N)} (t_0, \xi) \varphi_j (\xi)| \leq C_{j,N}
\label{K0}
\end{align}
 holds, where  $j
 \in \{0,1\}$ and $N \in \{0,1,2\}$.
\end{proposition}
By this proposition, $\zeta _j ^{(N)} (t_0,p) \varphi_j (p)$ can
be defined as a bounded operator on $L^2({\bf R}^n) $ through
the Fourier transform since
\begin{align*}
\left\|
\SCR{F}_1^{-1}\zeta _j ^{(N)} (t_0,\xi) \SCR{F}_1^{+1}\varphi_j(\xi) \psi_{0,j}
\right\|_{L^2({\bf R}^n)} &=  \left\| \zeta _j ^{(N)} (t_0, \xi)\varphi _j (\xi) \SCR{F}_1^{+1} \psi_{0,j}
\right\|_{L^2({\bf R}^n)} \\ &\leq C_{j,N} \left\| \psi_
 {0,j}\right\|_{L^2({\bf R}^n)}
\end{align*}
holds. It also follows that for any
fixed $t$, $\zeta
_j(t,p)$ can be defined on $\SCR{F}_1^{-1} C_0({\bf R}^n)$ since $\varphi_j (\xi)$ is independent of $t$ and satisfies $\zeta _j^{(N)} (t,p)
\varphi_j(p) \psi_{0,j} = \zeta _j^{(N)}(t,p) \psi_{0,j}$. The following proposition extends the domains of $\zeta _1(t,p)$ and $\zeta
_2 (t,p)$ from
$\SCR{F}_1^{-1}  C_0({\bf R}^n)$ to $H^{1/2} ({\bf R}^n)$ and $H^{-1/2}
({\bf R}^n)$, respectively.

\begin{proposition}\label{P1}
Suppose Assumption $\mathrm{ \bf (E1)}$ holds. Let $A(t,\xi)$ and $C(t,\xi)$ be equal to those defined in \eqref{8}. Then there exist $0< C_{0} < \infty$
 and $0 < C_{1} <  \infty$, independent of $t$ and $\xi$, such that
\begin{align}\label{9}
& \frac{Q(0, \xi)^{1/2}}{Q(t, \xi)^{1/2}} e^{-C_{0}}  \leq
 \left|
A(t, \xi)
\right| \leq \frac{Q(0, \xi)^{1/2}}{Q(t, \xi)^{1/2}} e^{C_{0}} ,  \\ & \label{9-2}
 \frac{1}{Q(t, \xi)^{1/2}Q(0, \xi)^{1/2}} e^{-C_{1}}  \leq
\left|
C(t, \xi)
\right| \leq \frac{1}{Q(t, \xi)^{1/2}Q(0, \xi)^{1/2}} e^{C_{1}}
\end{align}
hold.
\end{proposition}
\begin{proof}
For simplicity, we denote $Q(t,\xi) = Q(t)$ and $B(t, \xi) = B(t)$.
We only calculate the term $A(t ,\xi) \varphi (\xi)$; the term $C (t ,\xi)
 \varphi (\xi)$ can
be calculated in a similar manner.

By simple calculations, it follows that
\begin{align}\nn
& \int_0^{t} \frac{Q'(s)}{Q(s)} \sin ^2(B(s)) ds \\ & \quad =
\frac{1}{2} \left(
\LOG{Q(t)}- \LOG{Q(0)} - \int_0^t  \frac{Q'(s)}{Q(s)} \COS{2 B(s)} ds
\right).  \label{20}
\end{align}
Hence, to prove Proposition \ref{P1}, it suffices to show that the last term of the right-hand side of the above equation
 is uniformly bounded in $t$ and $\xi$.
Noting \eqref{t2}, we have
\begin{align} \nn
B'(s) &= Q(s) - Q'(s)Q(s)^{-1} \sin (2B(s))/2  \geq Q(s)/2 +
 \left(  Q(s)/2 - E_{0,0}/(2mc)
\right) \\  & \geq Q(s)/2 + \left( c |\xi + b(s)|  -
 E_{0,0}/(mc)\right) /2. \label{M0}
\end{align}
Next, we define
\begin{align*}
\Omega &:=
\left\{
s \in [0,t] \ :\  |\xi  +b (s)| \leq 2 E_{0,0}/(mc ^2 )
\right\}.
\end{align*}
Then by Assumption {\bf (E1)} and \eqref{t2}, we obtain that
\begin{align*}
\left|\int_{\Omega} \frac{Q'(s)}{2Q(s)} \cos (B(s)) ds \right| \leq
C \int_{\Omega} |b'(s)| ds  \leq C e_0
\end{align*}
is bounded and independent of $t$ and $\xi$. Conversely, on
 the region $[0,t] \backslash \Omega $, by \eqref{M0}, it always follows that
\begin{align} \label{M1}
B'(s) \geq Q(s)/2;
\end{align}
hence, it also follows that
\begin{align*}
& \int_{[0,t] \backslash \Omega } \frac{Q'(s)}{Q(s)} \cos (2B(s))
 ds \\ & =
\left[ \frac{Q'(s)}{2Q(s)B'(s)} \sin (2B(s))\right]_{\partial ([0,t]
 \backslash \Omega )}
 - \frac12 \int_{[0,t] \backslash \Omega }  z_1(s) \SIN{2B(s)} ds,
\end{align*}
where
\begin{align*}
 z_1(s)   &=
\frac{1}{(B'(s))^2 Q(s)^2} \left\{ Q''(s)Q(s)B'(s) -(Q'(s))^2 B'(s) -Q(s)Q'(s)B''(s)
\right\}
\\ &=   \frac{Q''(s)}{(B'(s))^2} - \frac{ 2(Q'(s))^2}{Q(s)(B'(s))^2}
+ \frac{(Q'(s))^2 }{2Q(s)^2 B'(s)} \cos (2B(s)) .
\end{align*}
Since \eqref{M1} holds and
\begin{align*}
|Q'(s)| \leq C |b'(s)|, \quad |Q''(s)| \leq C \left( |b''(s)|  +
 |b'(s)|^2 \right),
\end{align*}
it follows that on $[0,t] \backslash \Omega$,
\begin{align*}
|z_1 (s)| \leq C \left(
|b''(s)| Q(s)^{-2} + |b'(s)|^2Q(s)^{-2}
\right),
\end{align*}
where  $|Q(s)|^{-1} \leq C$ and $|b'(s)| = |E(s)| \leq
 E_{0,0}$. Hence, by Assumption {\bf (E1)},
\begin{align*}
\left|
\int_0^t \frac{Q'(s) \cos (2B(s))}{Q(s)} ds
 \right| &\leq C + Ce_0 + C \int_0^t \frac{|b'(s)|^2 + |b''(s)|}{c^2(\xi
 +b(s))^2 + (mc^2)^2} ds \\ &\leq C(1 +e_0 + e_1).
\end{align*}
Therefore, the proposition holds.
\end{proof}

By analyzing $\zeta _1$ and $\zeta _2$, we arrive at the following theorem.
\begin{theorem} \label{T0}
Let $\psi_0(t,x)$, $\psi _{0,0}$, and $\psi_{0,1}$ be equal to those defined
 in \eqref{1}. Suppose Assumption $\mathrm{ \bf (E1)}$ holds and that $\hat{\psi}_{0,0} \in C_0 (\bfR ^n)$ and
 $\hat{\psi}_{0,1} \in C_0 (\bfR ^n)$. Then for all $\theta
 \in \bfR $, there exists $0< C_{0,\theta }< \infty$ such that
\begin{align}\label{l1}
\left\| (L (0,p))^{\theta} {\psi} _0 (t,x) \right\|_{L^2(\bfR ^n)}
 \leq C_{0,\theta} |b(t)|^{(2\theta -1/2)}, \quad |t| \to \infty
\end{align}
holds. In particular,
\begin{align}\label{Q11}
\left\| \psi _0 (t,x) \right\|_{L^2(\bfR ^n)} \leq C_{0,0} |b(t)|^{- 1/2},
 \quad |t| \to \infty
\end{align}
holds, where $C_{0, \theta}$ is a constant depending only on the volume of the support
 of $\hat{\psi}_{0,0}$ and $\hat{\psi}_{0,1}$.
\end{theorem}
Solutions to \eqref{1} when the electric fields are independent of time have been investigated (see
Narozhnyi and Nikishov \cite{NN}, Tanji
\cite{Ta}, and \cite{Ve}); rotating electric fields were
investigated by Eliezer, Raicher, and Zigler \cite{ERZ}. However,
time-decay estimates \eqref{l1} and \eqref{Q11} have not been
considered.
\begin{proof}
On the support of $\hat{\psi}_{0,0}$ and $\hat{\psi}_{0,1}$, $Q(0,\xi)^{1/2}$ is
 bounded and
$$
C |b(t)|^{2} \leq L(t, \xi)^{1/2} \leq C |b(t)|^{2} $$ holds for $t \gg 1$. Thus, the inequality
\begin{align*}
\left\|
L(t,\xi)^{\theta}\zeta _j (t, \xi) \varphi _j (\xi) \hat{\psi}_{0,1} (t,
 \xi)
\right\|_{L^2(\bfR ^n)}
& \leq C \left\| L(t,\xi)^{ \theta - 1/4} L(0, \xi)^{1/4} \varphi_j (\xi)
 \hat{\psi}_j \right\|
\\ & \leq C|b(t)|^{(2 \theta -1/2)} \|
 \hat{\psi}_{0,1} (t, \xi) \|_{L^2(\bfR ^n)}
\end{align*}
holds from \eqref{9} and \eqref{9-2}, where $j \in \{0,1\}$. Therefore, Theorem \ref{T0}
 holds.
\end{proof}

\section{Proof of Theorem \ref{T2}}

In this section, we prove Theorem \ref{T2}. First, we decompose $U_{0,\alpha} (t)$ by using Hochstadt type
representations \eqref{L5}, \eqref{L6}, and \eqref{K3}. Then, by using
this factorization of $U_{0, \alpha} (t)$, we prove the stability and
instability properties.

\subsection{Factorization of $U_{0, \alpha} (t)$}
Noting the
definition of $U_{0,\alpha}
(t)$ (see \eqref{11}), $U_{0, \alpha} (t)$ can be factorized by
\begin{align} \nn
 U_{0,\alpha} (t) &:= K_{\alpha}^{1/2} U_{A_0} (t) (K_{\alpha}^{1/2})^{-1}  \\
 \nn  &= (e^{ib(t) \cdot x})_M \MAT{L(t,p)^{1/4-\alpha /2} & 0 \\ 0 &
 L(t,p)^{-1/4 - \alpha /2}} \\ & \quad \times \nn
\MAT{\zeta _0 (t,p) & \zeta _1 (t,p) \\ i \zeta _0'(t,p) & i \zeta _1
 '(t,p)} \MAT{L(0,p)^{-1/4+ \alpha /2} & 0 \\ 0 & L(0,p)^{1/4 + \alpha /2}}
\\ \nn
&= \MAT{ e^{ib(t) \cdot x} {\mathscr F}_1^{-1}
 {\mathscr L}_{\alpha} (t, \xi) & 0 \\ 0 &  e^{ib(t) \cdot x} {\mathscr F}_1^{-1}
 {\mathscr L}_{\alpha} (t, \xi) } \\  \quad & \quad \times\MAT{ {\mathscr G}_0(t,\xi)
 \cos (B(t,\xi)) & {\mathscr G} _1(t, \xi) \sin (D(t,\xi)) \\
 -i {\mathscr G}_0 (t, \xi) \sin (B(t,\xi)) & i {\mathscr G}_1 (t, \xi)\cos (D(t,\xi))
} \MAT{ {\mathscr F}_1^{+1} & 0 \\ 0 & {\mathscr F}_1^{+1}}, \label{AV}
\end{align}
where ${\mathscr L}_{\alpha} (t,\xi) = L(t,\xi)^{-
\alpha /2} L(0,\xi)^{ \alpha /2} $, $$
{\mathscr G} _0(t, \xi) = e^{\int_0^t (Q'(s,\xi) \cos (2B(s, \xi))
/(2Q(s, \xi))) ds}
$$ and $$ {\mathscr G}_1(t, \xi) = e^{-
\int _0^t (Q'(s, \xi) \sin (2D(s, \xi)) /(2 Q(s, \xi)) )
ds}.$$
This formula is a natural extension of the Avron-Herbst formula.

\subsection{Stability of $U_{0,0} (t)$ on $\SCR{H}$}
Here, we prove the first statement of Theorem \ref{T2}. Noting that ${\mathscr
F}_1^{-1} C_0({\bf
R}^n) \times {\mathscr F}_1^{-1}C_0({\bf R}^n)$ is dense on ${\mathscr H}$, every calculation is
done on $ \Phi _{0,0} \in {\mathscr F}_1^{-1}C_0({\bf R}^n) \times {\mathscr F}_1^{-1}C_0({\bf R}^n)$.
By \eqref{AV} with $\alpha =0$, together with the fact that
\begin{align}\label{L7}
e^{-C_0} \leq | {\mathscr G}_0(t, \xi)| \leq e^{C_0} , \quad e^{-C_1} \leq
 |{\mathscr G}_1 (t, \xi)| \leq e^{C_1},
\end{align}
holds by \eqref{9} and \eqref{9-2}, we have that there exists $C >0$
independent of $t$ and the support of $\Phi_{0,0}$ such that $\left\|
U_{0,0}(t) \Phi_{0,0}
\right\|_{\SCR{H}} \leq C \left\| \Phi _{0,0}\right\|_{\SCR{H}}$ holds. By the density argument, we also have
$\left\| U_{0,0}(t)\right\|_{{\mathscr B}({\mathscr H})} \leq C$. Next, we prove $\left\| U_{0,0}(t) \Phi_{0,0}
\right\|_{\SCR{H}} \geq C >0$. Letting $\Phi_{0,0} = \left( \phi_0, \phi_1
\right)^{\mathrm{T}}$, we have
\begin{align}\label{L9}
\left\|
U_{0,0} (t) \Phi_{0,0}
\right\|_{\SCR{H}}^2 &= \left\| \SCR{G} _0(t ,\xi) \hat{\phi}_0
\right\|^2_{L^2(\bfR ^n)} + \left\|
\SCR{G} _1 (t, \xi) \hat{\phi}_1
\right\|_{L^2(\bfR ^n)} ^2 \\ & \qquad - 2 \mathrm{Re} \left(
\SCR{G} _0(t , \xi) \SCR{G} _1 (t, \xi)\sin (B(t, \xi)-D (t, \xi)) \hat{\phi}_0, \hat{\phi}_1
\right)_{L^2(\bfR ^n)}. \nn
\end{align}
Using the fact that
\begin{align*}
\zeta _0 (t, \xi) \zeta _1'(t, \xi) - \zeta _0 '(t, \xi) \zeta _1 (t,
 \xi) = 1,
\end{align*}
we obtain
\begin{align}\label{L8}
\SCR{G} _0 (t , \xi)\SCR{G} _1(t, \xi) \cos (B(t, \xi)-D (t,\xi)) =1.
\end{align}
Inequalities \eqref{L7} and \eqref{L8} imply that for all
$t \in \bfR$ and $\xi \in \bfR ^n$, there exists $ 0 < \delta \leq 1$ such that
\begin{align*}
| \cos ( B(t, \xi) - D(t, \xi) ) | > \delta
\end{align*}
holds, i.e.,
$$
|\SIN{B(t, \xi) -D(t, \xi)}| < \sqrt{1- \delta ^2}
$$ holds. Using this inequality, \eqref{L7}, \eqref{L9}, and
\begin{align*}
\left\|U_{0,0} (t) \Phi _{0,0} \right\|_{\SCR{H}}^2 & \geq (1- \sqrt{1-
\delta ^2}) \left( \left\| \SCR{G} _0 (t ,\xi) \hat{\phi}_0
\right\|^2_{L^2({\bf R}^n)} + \left\|
\SCR{G}_1 (t, \xi) \hat{\phi}_1 \right\|^2_{L^2({\bf R}^n)} \right) \\ &
 \geq (1- \sqrt{1- \delta ^2}) \left(\min \{ e^{-2C_0}, e^{-2C_1}\} \right) \left\| \Phi_{0,0} \right\|^2_{\SCR{H}},
\end{align*}
we obtain Theorem \ref{T2}.
\subsection{Instability of $U_{0, \alpha}
  (t)$, $\alpha \neq 0$, on $\SCR{H}$}
We now complete the proof of Theorem \ref{T2}. By \eqref{AV}, for $\Psi _0 =(\psi_0, \psi_1)^{\mathrm{T}}\in \SCR{F}^{-1}C_0 (\bfR ^n) \times
\SCR{F}^{-1}C_0(\bfR ^n)$, simple calculations show that
\begin{align*}
\left\|
U_{0, \alpha} (t) \Psi_{0}
\right\|_{\SCR{H}}^2 &= \left\|  \sigma_{0, \alpha}(t, \xi)\hat{\psi}_0 \right\|^2_{L^2(\bfR ^n)} +  \left\|  \sigma_{1, \alpha}(t, \xi)\hat{\psi}_1
\right\|^2_{L^2(\bfR ^n)} \\ & \quad
-2 \mathrm{Re} (\sigma_{0, \alpha } (t, \xi) \sigma _{1, \alpha } (t, \xi) \sin (B(t, \xi) -D(t, \xi))\hat{\psi}_0, \hat{\psi} _1)_{L^2(\bfR ^n)}
\end{align*}
holds, where $\sigma _{0, \alpha}$ and $\sigma _{1, \alpha}$ are defined
by
\begin{align*}
\sigma _{0, \alpha} (t, \xi) = \SCR{G}_0(t, \xi) \SCR{L}_{\alpha} (t,
 \xi), \quad
\sigma _{1, \alpha} (t, \xi) = \SCR{G}_1(t, \xi) \SCR{L}_{\alpha} (t,
 \xi).
\end{align*}
In the same way as the proof of the stability of $U_{0,0}(t,0)$, we have that there exist $c_{00} >0$ and $\delta _{00} > 0$ such that
\begin{align*}
\left\|
U_{0, \alpha } (t) \Psi_{0}
\right\|_{\SCR{H}}^2
\begin{cases}
\leq  & c_{00} \left(\left\|  \sigma_{0, \alpha}(t, \xi)\hat{\psi}_0
 \right\|^2_{L^2(\bfR ^n)} +  \left\|  \sigma_{1, \alpha }(t, \xi)\hat{\psi}_1 \right\|^2_{L^2(\bfR ^n)}\right), \\
\geq  & \delta_{00}\left(\left\|  \sigma_{0, \alpha }(t, \xi)\hat{\psi}_0
 \right\|^2_{L^2(\bfR ^n)} +  \left\|  \sigma_{1, \alpha }(t, \xi)\hat{\psi}_1 \right\|^2_{L^2(\bfR ^n)}\right),
\end{cases}
\end{align*}
holds. On the other hand, by \eqref{9} and \eqref{9-2}, note that for $j \in \{0,1\}$, there exist
$0 < \tilde{c}_j < \tilde{C}_j$ such that
\begin{align*}
&
\tilde{c}_j \left\|
\SCR{L}_{\alpha}(t, \xi) \hat{\psi}_j
\right\|_{L^2({\bf R}^n)} ^2 \leq \left\|
\sigma_{j, \alpha}(t,\xi) \hat{\psi }_j
\right\|^2_{L^2(\bfR ^n)} \leq \tilde{C}_j  \left\|
\SCR{L}_{\alpha} (t,\xi) \hat{\psi}_j
\right\|_{L^2({\bf R}^n)} ^2
\end{align*}
holds, where $\SCR{L}_{\alpha} (t,\xi) :=
(L(t,\xi))^{- \alpha /2} (L(0, \xi))^{ \alpha /2}$. Clearly, $L(t, \xi)=
c^2(\xi + b(t))^2 + (mc^2)^2 \to \infty $ as $t \to \infty$ holds on $C_0({\bf R}^n) \times C_0({\bf R}^n)$; hence, it follows that
for $t \to \infty$,
\begin{align*}
\left\|
\sigma_{j, \alpha } (t, \xi) \hat{\psi}_0
\right\|^2_{L^2(\bfR ^n)} \to
\begin{cases}
0, & \mbox{ if } \alpha >0, \\
\infty, & \mbox{ if } \alpha <0,
\end{cases}
\end{align*}
holds.

redpajama_set_name: RedPajamaArXiv


Kyrie Irving's "Uncle Drew" movie will be making its way to theaters on June 29th and Nike will be dropping a special colorway of the Nike Kyrie 4 to coincide with the release of the film.
The Nike Kyrie 4 Uncle Drew is dominated by White all throughout the upper with a Deep Royal Blue hitting the Swoosh logos on the side panels. A white midsole witj colorful speckling and a gum rubber outsole round out the look.
Be on the lookout for the Nike Kyrie 4 Uncle Drew to release on in full family sizes on June 23rd for $120. Stay tuned to Kicks On Fire for more images as soon as they are made available.


text:
Norsk Hydro and Golar Power sign MOU to develop LNG terminal in Pará
Jul 23. 2020
Norwegian Norsk Hydro and electric company Golar Power signed a memorandum of understanding aimed at developing a liquefied natural gas (LNG) terminal in Barcarena, Pará.
The project aims to allow Golar Power to supply gas to the Alunorte aluminum refinery, a Norsk unit near the port of Vila do Conde, also in Barcarena.
Alunorte would be the first operational customer for a floating LNG regasification unit (or FSRU) that Golar Power plans to put into operation in the region in the first half of 2022, the company said.
"The conclusion of final agreements with Norsk Hydro will therefore be an important step towards the start of investments, which should take place in the next 4 to 6 months," added in a note to Golar Power, a joint venture between Golar LNG and investment firm Stonepeak Infrastructure Partners.
In addition to Alunorte, the LNG terminal will also supply the fuel to supply a 605 megawatt thermal power plant that a Golar Power subsidiary will build in the region and which already has contracts for the sale of production for 25 years, according to the company.
Source: Reuters, 23/0272020
Brazil, Gas
EMGS loss widens on halved revenues
Havila bags three-year job for subsea vessel in Brazil

redpajama_set_name: RedPajamaCommonCrawl


text:
New Year Honours 2017: leading scientists recognised in list
Appointees in the Queen's New Year's Honours List include the researchers behind DNA fingerprinting and the most effective therapy for malaria
Hilary Lamb
Twitter: @_HilaryLamb
Prominent figures in science and technology have been acknowledged in the Queen's New Year Honours, alongside sportspeople, entrepreneurs and philanthropists.
The 2017 New Year's Honours List has been billed as the most diverse and inclusive so far, with more than half of the honours going to women and more ethnic minority appointees than ever before.
Sir Cyril Chantler, the paediatric kidney specialist who has held numerous leading roles in healthcare, has been appointed Knight Grand Cross of the British Empire. He is the former chair of Great Ormond Street Hospital, former pro vice-chancellor of the University of London and is now emeritus professor of Guy's, King's and St Thomas's Medical School. He is known for having led an influential independent review into tobacco packaging.
The "founding father of genetic profiling", Sir Alec Jeffreys, has been appointed a Companion of Honour. Currently emeritus professor at the University of Leicester, he is celebrated for having developed DNA fingerprinting and investigating how DNA is affected by environment impacts.
Among those appointed Knight or Dame Commander of the Order of St Michael and St George are Jane Francis, who has been director of the British Antarctic Survey since 2013, and Nicholas White, chairman of the Wellcome Trust and professor of tropical medicine at the University of Oxford and Mahidol University, Thailand. His achievements include leading clinical trials that established the most effective treatment yet for malaria; work that has saved hundreds of thousands of lives.
The list also includes a damehood for Ottoline Leyser, director of the Sainsbury Laboratory at the University of Cambridge, and knighthoods for James Smith, director of research at the Francis Crick Institute; Barry Ife, principal of Guildhall School of Music and Drama; Antony Beevor, author and visiting professor of history at Birkbeck, University of London; and Nicholas Black, professor of health services research at the London School of Hygiene and Tropical Medicine.
Steve West, vice-chancellor of the University of the West of England, was appointed CBE. Janice Kay, provost of the University of Exeter, and Cecilia Anim, president of the Royal College of Nursing, received the same honour.
Among those appointed OBE was Joe Baden, an academic at Goldsmiths, University of London who left school at 13 with no qualifications and faced a prison sentence before becoming determined to go to university. At Goldsmiths he founded the Open Book project, which works with people with addictions or from criminal backgrounds to encourage them to take up education.
New Year Honours 2017
Among those appointed Knight Grand Cross of the Order of the British Empire:
Sir Cyril Chantler, emeritus professor of Guy's, King's and St Thomas's Medical School. For services to leadership in healthcare.
Among those awarded Order of the Companion of Honour:
Sir Alec Jeffreys, emeritus professor, University of Leicester. For services to medical research and society.
Among those appointed Knight/ Dame Commander of the Order of St Michael and St George:
Professor Jane Francis, director, British Antarctic Survey. For services to UK polar science and diplomacy.
Professor Nicholas White, chairman, Wellcome Trust and professor of tropical medicine, University of Oxford. For services to tropical medicine and global health.
Among those appointed Companion of the Order of St Michael and St George:
Professor William Aspinall, Cabot professor in natural hazards and risk science, University of Bristol. For services to the government and community in Montserrat.
Among those given knighthoods:
Professor Shankar Balasubramanian, Herchel Smith professor of medicinal chemistry, University of Cambridge. For services to science and medicine.
Antony Beevor, visiting professor of history at Birkbeck, University of London. For services in support of armed forces professional development.
Professor Nicholas Black, professor of health services research, London School of Hygiene and Tropical Medicine. For services to healthcare research.
Professor Barry Ife, former principal, Guildhall School of Music and Drama. For services to performing arts education.
Professor John McCanny, professor of microelectronics engineering, Queen's University Belfast. For services to higher education and economic development.
Dr James Smith, director of research, Francis Crick Institute. For services to medical research and science education.
Among those made dames:
Professor Elizabeth Anionwu, emeritus professor of nursing, University of West London. For services to nursing and the Mary Seacole Statue Appeal.
Professor Amanda Fisher, director of the Medical Research Council Clinical Sciences Centre, Imperial College London. For services to medical research and public understanding of science.
Professor Ottoline Leyser, director, the Sainsbury Laboratory, University of Cambridge. For services to plant science, science in society and equality and diversity in science.
Professor Caroline Watkins, professor of stroke and older people's care, University of Central Lancashire. For services to nursing and older people's care.
Among those appointed CBE:
Cecilia Anim, president of the Royal College of Nursing. For services to women's health.
Professor Polina Bayvel, professor of optical communications and networks, University College London. For services to engineering.
Professor Kamaldeep Bhui, professor of cultural psychiatry and epidemiology, Queen Mary University of London. For services to mental health research.
Professor Richard Burdett, professor of architecture and urban studies, London School of Economics. For services to urban planning and design.
Professor Jack Cuzick, director, Wolfson Institute of Preventive Medicine and head, Centre for Cancer Prevention. For services to cancer prevention and screening.
Professor Susan Deacon, chair, Institute of Directors in Scotland and assistant principle external relations, University of Edinburgh. For services to business, education and public service.
Professor Michael Depledge, chair of environment and human health, University of Exeter. For services to the environment and human health.
Professor Peter Goodhew, emeritus professor of engineering, University of Liverpool. For services to engineering and education.
Professor Christopher Haslett, Sir John Crofton professor of respiratory medicine and director, Queen's Medical Research Institute, University of Edinburgh. For services to medical research.
Professor Stephen Inglis, former director, National Institute for Biological Standards and Control. For services to health protection.
Professor Janice Kay, provost and senior deputy vice-chancellor, University of Exeter. For services to higher education.
Professor Nicola Lacey, professor of law, gender and social policy, London School of Economics. For services to law, justice and gender politics.
Professor Paul Martin, depute principal, University of the West of Scotland. For services to healthcare and education.
Professor Jill Pell, Henry Mechan professor of public health, University of Glasgow. For services to public health research.
Professor John Pyle, professor of chemistry, University of Cambridge. For services to atmospheric chemistry and environmental science.
Luke Rittner, chief executive, Royal Academy of Dance. For services to the arts, particularly dance.
Professor Emeritus Carol Smart, professor of sociology, University of Manchester, and co-director, Morgan Centre for the Study of Relationships and Personal Life. For services to the social sciences.
Professor John Spencer, professor emeritus of law, University of Cambridge. For services to the reform of law concerning child witnesses.
Professor Anita Thapar, clinical professor, Institute of Psychological Medicine and Clinical Neurosciences, Cardiff University. For services to child and adolescent psychiatry.
Professor Steven West, vice-chancellor, University of the West of England. For services to higher education.
Professor Guang-Zhong Yang, director, Hamlyn Centre for Robotic Surgery. For services to biomedical engineering.
Among those appointed OBE:
Joe Baden, founder, Open Book Project. For services to higher education.
Professor Claire Callender, professor of higher education policy, Birkbeck, University of London, and professor of higher education, UCL Institute of Education. For services to education.
Christine Craik, director of occupational therapy, Brunel University. For services to occupational therapy, particularly mental health services and education.
Professor Valentine Cunningham, former professor of English language and literature, University of Oxford. For services to scholarship and understanding of the humanities.
Professor Mary Dunning, former governor, Tyne Metropolitan College. For services to further and higher education and the community in the North East of England.
Professor Aminatta Forna, professor of creative writing at Bath Spa University. For services to literature.
Professor Simon Frith, Tovey professor of music, University of Edinburgh. For services to higher education and popular music.
Professor John Furlong, emeritus professor of education, University of Oxford. For services to research in education and government.
Professor Susan Hunston, professor of English language, University of Birmingham. For services to higher education and applied linguistics.
Professor Mary Lumsden, professor of obstetrics and gynaecology, University of Glasgow. For services to women's health.
Professor Ghulam Mufti, professor of haemato-oncology, King's College London. For services to haematological medicine.
Professor James Neil, professor of virology and molecular oncology, University of Glasgow. For services to the advancement of biomedical sciences.
Professor Susan Rhind, professor and chair of veterinary education, the Royal (Dick) School of Veterinary Studies. For services to veterinary education.
Professor Rosalind Richards, head of the International Study and Language Institute, University of Reading. For services to language support for international education.
Professor Tess Ridge, professor of social policy, University of Bath. For services to the social sciences.
Professor Robin Sengupta, chairman, Institute of Neuroscience, Kolkata. For services to people with neurological disease in the UK and India; and to the UK-India relationship.
Professor David Vaughan, glaciologist, British Antarctic Survey. For services to glaciology.
Professor Sally Wheeler, professor of law and head of the School of Law, Queen's University Belfast. For services to higher education in Northern Ireland.
Among those appointed MBE:
Professor Deborah Bowman, professor of medical ethics, St George's University of London. For services to medical ethics.
Martin Devenish, alumni support, University of Birmingham. For services to higher education and voluntary service to educational development overseas.
Professor Andrew George, chair, Hammersmith Hospital Research Ethics Committee. For services to research participants and the ethical governance of clinical research.
Steven Grant, assistant director HR, University of Manchester. For services to unemployed people in greater Manchester.
Professor Kamila Hawthorne, clinical professor of medical education and associate dean for medicine, University of Surrey. For services to general practice.
Kathleen Hood, former head of widening participation, University of Edinburgh. For services to widening participation.
Professor Wen Guo Jiang, professor of medicine, Cardiff University. For services to international cancer research.
Sara Kibel, governor, Westminster Adult Education Service. For services to adult education.
Michael McAlister, deputy director and head of curriculum and business development, South West College. For services to further education in Northern Ireland.
Sarah Murray, senior lecturer in dentistry, Queen Mary University of London. For services to oral health.
Dr Terence Pennington, honorary research fellow, Kew. For voluntary service to plant sciences and conservation, particularly in Latin America.
Professor Jonathan Stephens, former director of music and music education, University of Aberdeen. For services to music education.
Denise Tully, bequeathal officer, University of Liverpool. For services to medical education and research.
Willie Walters, programme director for fashion, Central Saint Martins, University of the Arts London. For services to higher education, fashion and the creative industries.
Dr Premila Webster, director of public health education and training, Nuffield Department of Population Health, University of Oxford. For services to public health.
Among those awarded the British Empire Medal:
David Emley, former technician, Keele University. For services to higher education and natural history in Staffordshire.
Professor Peter Heasman, volunteer member, NHS Research Ethics Committees. For services to providing ethical review and support to researchers.
Ellen Scott, former senior administrator, University of Edinburgh. For services to research and the community in Gorebridge, Midlothian.
Hillsborough researcher turns down OBE
Phil Scraton says he "could not receive an honour on the recommendation of those who remained unresponsive" to campaigns for justice
By Hilary Lamb
Reader's comments (2)
#1 Submitted by Charles Hedges on December 31, 2016 - 2:00pm
Note that on your sidebar for this article you wrongly assign the Damehood to Henrietta Leyser, medieval historian and mother of Dame Ottoline Leyser.
#2 Submitted by DrHowardFredrics on January 1, 2017 - 3:58pm
New Year honours 'reward failure' as civil servants behind Whitehall fiascos are given awards Civil servants behind some of the year's biggest controversies have been "rewarded for failure" in the Queen's New Year honours, critics said, despite Theresa May's insistence that the system will be reformed. Mark Lowcock, the Permanent Secretary at the Department for International Development, is knighted despite repeated controversies over the allocation of £12 billion per year in foreign aid. Others who have been honoured include senior staff at HM Revenue and Customs, which has driven taxpayers to distraction with its poor service; the boss of the Driver and Vehicle Licensing Agency and senior staff at the Border Force.


A few little HTML tricks that will help users format their own additions to the WiccaPlus app.
</table> and this tag tells your code the table is done. These two tags will wrap around the meat of your table.
</tr> and to end that row.
</td> and where that cell ends.
Together they would look like this- ignore the text in parentheses.
This text would appear in your first cell of your first row.
To make bigger tables you just add more cells and rows, remember that cells go inside rows, so you have to close your tags in the reverse order that you wrote them. Also note you don't have to close each tag on it's own line, it's written that way here for easier comprehension.
This will be your table Header!
Because "column span" is modifying your table header tag, but isn't a separate tag on its own, you only have to close <th>.
To add a border to separate the cells in your table you modify your table tag.
This adds a 1 pixel wide border around all of your cells. There's ton of resources online to learn more HTML but I thought a quick crash course in tables would be handy. I'll follow up with how to add a clickable table of contents that will let you quickly navigate a long page.
Need to navigate a long page? A table of contents at the top will help you jump down to where you want to go without scrolling.
That tag sets a sort of anchor to the word Fire in your text, and names the anchor "fire". Back at your table of contents near the top of your page, you add a link that goes to your anchor, rather than a webpage.
This turns the word Fire in your list into a link that will navigate to your anchor on the same page. Note that even thugh the anchor is named "fire" the link to it requires a # symbol as well.
This is awesome, thank you so much Scorpius!
One forum-related tip I'll add is to make posted-code easier to read, there is a formatting-option in the forum itself. It's above the 'Your Message' box, towards the right, and looks like a bold # symbol.
(2014.03.11, 15.27)Scorpius Wrote: Need to navigate a long page? A table of contents at the top will help you jump down to where you want to go without scrolling.
OOH! OOH!! I love jump-to links!!
I have trouble with them showing up where they're supposed to though.
Sometimes the jump will show up way below and I have to place the line destination 4 lines above my target.


Little did we know that there are instances in our life where we're too caught up with so many things that it stresses us out. Because of these underlying factor, we're now battling with changes in mood, irritability, restlessness, and intermittent anxiety. And without us being aware of what is happening, these will then have a huge effect on how we interact with other people, especially our loved ones.
The good news is, there is a better way to help us get over all these. Scientifically formulated with a precise balance of high-quality ingredients, Elimidrol boosts your mood and relieves intermittent anxiety, restlessness, and irritability. Elimidrol is the future of life-enhancing supplements. Life can be difficult to navigate when anxiety and unconstructive moods take the wheel; Elimidrol is there to help you regain control, so you can make every moment count.
Elimidrol is a scientifically formulated support supplement engineered to positively impact your daily life and help battle everyday stresses. Elimidrol promotes positive mood, comfort, calmness, and overall well-being, and provides a sense of mental and physical balance. Produced with high-quality life-enhancing ingredients, Elimidrol can be used daily or for occasional support.
Elimidrol knows what counts. Life can be hectic, and too often intermittent anxiety, low mood, and restlessness become overwhelming and can cause you to lose sight of what is important. Designed to go beyond just a mood enhancement product, Elimidrol is there to give you the support you can count on to make the most of your day.
The one of a kind combination of key ingredients, herbs, amino acids, minerals, and supporting nutrients were carefully chosen for their life-enhancing qualities and blended into a formula with the power to combat bouts of anxiety and improve overall mood. The dosage of these high-quality ingredients are precisely calculated to ensure safety and efficacy, so you can enjoy your day and focus on what matters most to you. Elimidrol is there to help you make every moment, of every day the best it can possibly be.
Elimidrol is designed to work for you. The powerful ingredients are formulated into a powder, rather than a tablet or capsule, offering the flexibility to control your dosage, day to day. Available in Daytime and Nighttime formulas, Elimidrol is there when you need it most.
The Elimidrol Daytime Formula is engineered to keep you in a positive mood and relieve anxiety to give you the support you need to meet your day's potential.
The Elimidrol Nighttime Formula offers similar benefits, but with an emphasis on relaxation and restful sleep, so you can wake up refreshed and eager to tackle your day.
The Elimidrol Daytime and Elimidrol Nighttime formulas can be used together for day & night support.
Elimidrol is an advanced mood enhancement and anxiety relief supplement that provides the mental and physical balance you deserve. Your moments in life are important—don't settle for anything less than a product that guarantees satisfaction. Elimidrol stands behind the product with a Money Back Guarantee, because what truly matters is that the product works for YOU.
Elimidrol knows what counts - Every Moment of Every Day.


Ethan is definitely a hand full. The sweetest boy you'll ever meet, but he just goes goes goes leaving destruction in his path. And Aiden is a baby that loves to be held and talked to which always makes it difficult to stay on Ethan's heels.
ME - Hello, how are things going?
ME - Are you sure you're okay?
CAKIE - (with a hint of panic and desperation) Yup, ummm, where do I find the stuff for Ethan to get in the pool?
CAKIE - (cutting me off) Okay great...Ethan, Cakie said NO...things are great - don't rush home. Need anything else?
ME - Can you check to see if we have any more Little Swimmers diapers? I think you said you saw some left up here from before. If not, I'll run back in to the store and grab some and we'll be on our way home.
CAKIE - Sure, hold on...you stay here Ethan, I'll be right back.
I hear her put down the phone (which is another one of Ethan's favorite things to play with) and leave the room, shutting the door to hopefully keep him contained as she's checking in the other room for the diapers, with Aiden on her hip. I hear the pitter patter of Ethan's feet getting closer to the phone, and then his chatter and some static as he's playing with the phone, probably upside down. I attempt to instruct him to put it down - but that doesn't even work when I'm visible to him, like he's going to listen to me when he can't even see me. It comes and goes and I'm also hearing what sounds like certain destruction of other objects he is not allowed to play with when finally Cakie re-enters in the background. I hear a few expletives at whatever Ethan has destroyed and she gets back on the phone, cheerily saying "Nope, didn't find them, got to go!"
I'm now a little nervous. Maybe it was a bad idea to leave so hastily. I should have been more organized and helped her find the things she would need before I ran out the door.
Luckily, I get home to find two happy babies. Cakie's holding Aiden and Ethan is waddling around in his swim trunks. Wait a second, not his swim trunks...Aiden's swim trunks!!! Somehow, she managed to squeeze a 16 month old into the wrong trunks - a size 0-3 months! He's completely unaware, and raring to get in the pool despite the fact that the elastic waistband is cutting off his circulation. I then take Aiden to change his diaper, thinking she probably didn't have a free minute to do this with chasing after Ethan. But I was wrong, she indeed did find the time. It wasn't the fact that Aiden was dry that gave it away, it was the fact that he had on Ethan's diaper (a size 4 - Aiden wears a size 1-2).
"Hey, they're alive and walking right?" Cakie said defending herself as we all had a good laugh. I can probably scratch her off my list of "babysitters crazy enough to offer to watch my kids". Regardless, I enjoyed my time out with my mom - so THANK YOU CAKIE!
Taryn - are you sure you really wanted to post this for all potential babysitters to see?! Almost scared me off but then realized that kids and babysitters do survive -- especially if there is a sense of humor. Loved the account.


text:
Saoussen Maalej () est une actrice tunisienne connue pour ses rôles dans des séries télévisées et des courts métrages. Elle est également animatrice de radio.

Elle a été mariée au journaliste tunisien Ilyes Gharbi.

Filmographie

Cinéma

Longs métrages
 2003 : Le Soleil assassiné d'Abdelkrim Bahloul
 2004 :
 La Villa de Mohamed Damak : Alia (sœur de Hédi)
 Parole d'hommes de  : Olfa
 2006 : Fleur d'oubli de Salma Baccar
 2013 :
 Jeudi après-midi de Mohamed Damak
 Affreux, cupides et stupides (Hezz Ya Wezz) d'Ibrahim Letaïef : Elvis
 2015 : Lala Zoubeïda Ouaness de Yahia Mouzahem : Zoubeïda Ouaness
 2018 :
 Regarde-moi de Nejib Belkadhi : Khadija
 Le Pardon de Najwa Limam Slama : Sofia

Courts métrages

Télévision

Séries

Téléfilms
 2003 : Khota Fawka Assahab d'Abdellatif Ben Ammar
 2009 : Choufli Hal d'Abdelkader Jerbi : Mariem - Mimi

Émissions
Saoussen Maalej débute à la télévision dans l'émission Chams Alik sur Canal+ Horizons : elle y est animatrice auprès de Nejib Belkadhi, de 1999 à 2001, année où l'émission s'interrompt en raison de la fermeture de la chaîne. Elle est ensuite animatrice dans le talk-show quotidien Ahla Snine dont Ilyes Gharbi est le rédacteur en chef, diffusée sur Canal 21. Puis, de 2009 à 2010, elle officie dans l'émission Ness Nessma sur Nessma aux côtés de l'animateur Fawez Ben Tmessek ; elle y interprète plusieurs personnages fictifs tels que Sassia, Anissa Poussette et Madama Gabsi dans des chroniques satiriques et parodiques. Certains de ces sketchs font cependant polémique.

Théâtre
 2005 : État civil d'Atef Ben Hassine
 2013 : Zanket Ensé, adaptation tunisienne de la pièce Le Clan des divorcées sur une mise en scène de Sami Montacer

Notes et références

Liens externes

Actrice tunisienne de cinéma
Actrice tunisienne de télévision
Actrice tunisienne de théâtre
Animatrice tunisienne de télévision
Date de naissance non renseignée (XXe siècle)
Lieu de naissance inconnu

redpajama_set_name: RedPajamaWikipedia


We provide University of Pennsylvania undergraduates with the chance to develop data analytics skills, explore career paths in the field of data analytics, network with professionals in data-driven industries, and be a member of a data-focused community at Penn.
In our Wharton Analytics Fellows branch, we partner with our clients to uncover, test, and scale analytical insights that can drive decision-making across industries and functions. By leveraging existing data resources to launch analytical pilots at no cost to our sponsoring organizations, we offer our clients a risk-free opportunity to capture value using the latest analytical innovations from the Wharton Customer Analytics Initiative.
WUDAC's Data@Nite Series won best New Student Engagement in 2017-2018. The 6 events were able to reach over 150 students. Undergraduates with no experience in analytics were able to come away knowing how to analyze datasets using SQL, R, Python, Tableau, and machine learning.


Brava Point Of Sale is another local business providing a feature loaded "all in the cloud" point of sale for retailers. With technology firms we always try to go as "clean" as possible to enhance the brand. The site lists all of Brava's many features with slide shows of screen shots from the actual interface. In addition the mobile version allows the user to slide through the same features with a mobile gallery.
Rosewood is a local San Diego art and consignment boutique. The owner approached us with a request for an eCommerce online store with a twist . Naturally we used the "rosewood" web font thought the site giving it that "retro old west yet contemporary" look. On this eCommerce project we modified the open court system with a blog and add a custom module that allows adding artists on one section and associating them with their art for sale. So while reading an article on a specific local artist you can visit his page and choose the add his products to your cart.
New Planet Moving, a Las Vegas moving and storage company providing local and long distance relocation services. The owner wanted to focus on the reviews and feedback the company receives from clients due to the facts that most people associate trust with a man who places your entire home on a truck and drives away. So we developed a new content type allowing him to not only add testimonials, but actually upload the letter sent by that client as a show of trust. In addition the site offer loads of useful content about preparing for a move and links for acquiring the proper insurance. In addition we made a mobile friendly site to allow prospective clients to request a quote for their coming move.
Washtek is an Escondido based company managing community on-premise laundry rooms. Obviously the main idea was to first explain to visitors what that means. So we designed a series of layered slideshow sliders showing the many aspects of the business. Washtek is a prefect example of a business that was not visible and now is. The previous website was designed using all flash which made it invisible for about 50% of today's traffic. The new site features HTML5 combined with j Query sliders and is responsive for any device.
Grande Laundry Place is another company from the founders of Washtek. Here in addition to taking the site away from flash and mobile invisibility there was a need to modify the WordPress CMS so that the owners may edit their locations. This allows Grande Laundry site administrators to specify the amenities in each location and add images. The specific URL for each location is now search engine friendly and local Escondido customers can find the branch closest to them.
Topical Stamp Albums founder Sergio is trying, as he is putting it "to revive a lost art". The idea he had was to build an online store that is handy for those stamp album collectors who wish to by topic and in bulk. We had to create a few modifications to the Open Cart system to allow that but we ended up with the most realistic Ecommerce interface that a stamp collector has today for acquiring stamp pages by the topic. The site is, like any other design we create responsive so you can also order stamp pages from your phone!
Jorge Garza, a local Poway artist gave us a free hand in designing his online gallery of works. What came out is a beautiful full screen portfolio site showing his art in every device. The site is managed with WordPress and allows Jorge to optimize his images before uploading to a portfolio for a faster user experience. The portfolio design is responsive and looks sleek in tablets and smart phones as well as on desktops.
When the first original Israeli grill in San Diego opened we felt obligated to take on the project and show local Hillcrest residents what they were missing all this time. High resolution images break through and let you smell the fresh bread as it bakes. The site administrators have a widget allowing them to upload images of the dishes to a featured section on the home page. In this case bringing a new type of cuisine required us to focus on more images and samples of what the food is about.
At Martin Plumbing, they do full service, including HVAC in the San Diego area. They wanted to be able to manage and add special offers & coupons to their website for clients, including being able to add things like testimonials and online contact forms, etc. We kept the new layout design in-unity with their already developed brand and logo. They're now able to manage things on their end and are 100% satisfied with the new build.
Manhattan shades is a New York City contractor who specializes in window treatments. The site was to resemble the different styles this specific business offers. Most of all the business owner wanted an easy way to update his latest projects with images and video which obviously did wonders for the sites SEO and ranking. This shows how updating your site with latest works can increase conversion through improved ranking and gaining visitors' trust through example.


text:
TLC Brought a Backyard Boogie Vibe to Irving With Their CrazySexyCool Tour Stop
Aria Bell October 6, 2021 2:27PM
TLC got comfortable with fans on their Tuesday night show, and why shouldn't they? They've only known them for 30 years. Jamie Ford/ Live Nation
TLC has made being crazy, sexy and cool look easy for nearly three decades. It's been 27 years since their iconic album CrazySexyCool climbed the charts, making them one of America's top-selling female groups of all time. Now they're back on the road to commemorate their diamond-certified album with fans such as those who showed up ready to party with the band Tuesday night at the Pavilion at Toyota Music Factory.
In June, TLC surprised fans by announcing their national "Celebration of CrazySexyCool" fall tour would be kicking off in early September. Initially, the 18-city tour was set to coincide with the album's 25-year anniversary mark but had to be put on hold indefinitely due to COVID-19. After a year and half of uncertainty, the group took to the road with an even greater appreciation for being able to perform again.
Ahead of their Irving show, we spoke with TLC's Chilli over the phone while she was on the road. The singer talked about what this tour means to the group, how it feels to perform following the pandemic, how CrazySexyCool skyrocketed groundbreaking careers and what's up next for the girl group.
It's no secret T-Boz's raspy voice, Chilli's smooth high notes and Left Eye's raw, confident raps — along with their "prissy tomboy" style — set the group apart from the many clone groups and one-hit wonders of the '90s. Chilli says they've remained relevant for nearly three decades because their lyrical content resonates with their fans.
"None of our songs are preachy, but they do make you think, and we're happy people of all ages can relate to it," Chilli said. "We all honestly go through the same things, some more than others, and it's something very powerful about music when you're going through good or bad times."
After TLC's debut in 1992, their following grew rapidly as fans fell in love with their "girl from around the way" vibe, empowering music and iconic fashion trends that redefined what sexy can look like. When their biopic CrazySexyCool: The TLC Story first aired on VH1 in 2013, they gained a whole new generation of fans. This was a full-circle moment for Chilli, who always believed the group's impact would be huge.
"Not from a narcissist place at all, but I told them there was going to be a movie about us one day after looking at our pictures during photoshoots," she said. "And this was before half the stories you saw even happened. They didn't believe me, but I knew something was special about us."
The Celebration of CrazySexyCool tour honors the work that changed the trajectory of their music careers and proved the naysayers wrong.
"This album showed that we weren't just a fad but that we had the chops and talent to stay," Chilli said. "It really let you see our chemistry as a group and how we worked so well together."
The band is anxiously anticipating the celebration of their 30-year anniversary milestone, but in the meantime, they are working on two projects that will include new TLC music and working on a two-hour documentary called Biography: TLC that will debut on A&E next year.
"It's such a blessing to still have fans that still tell us how our music touches them," Chilli said. "Just know when you see me smiling on stage, Chilli is just happy to be on it."
The pandemic is still haunting live music fans. About 24 hours ahead of TLC's Irving concert, promoter Live Nation notified all ticket holders that they'd need to present a vaccination card or negative COVID test results from within the last 72 hours to enter the venue. Some ticket holders who missed the memo stood shocked and frustrated outside of the entrance. Some said they were vaccinated but left their card at home, while fans who weren't vaccinated pondered whether to pay $40 to take a rapid COVID test offered onsite or else skip the show.
The concert date had already been pushed back from Sunday, Oct. 3, due to minor health concerns related to singer T-Boz, so the venue wasn't entirely packed out.
T-Boz soothed the crowd with her famous vocals.
Jamie Ford/ Live Nation
After openers Bone Thugs-N-Harmony warmed up the crowd with classics such as "1st of tha Month," "Thuggish Ruggish Bone," "Tha Crossroads" and more, TLC's DJ Dubz hit the stage around 8:45 p.m. and got the anxious crowd out of their seats while the band got in place.
Fifteen minutes later, pictures of the group that included late group member Lisa "Left Eye" Lopes — who died in a car crash in 2002 — appeared on all stage screens as TLC made its entrance from the top of the stage. T-Boz and Chilli kicked off the show in orange and maroon silk pajamas, performing their No. 1 single, "Creep." T-Boz's seductive voice soothed the crowd as seemingly every audience member sang along.
The pair followed up with "Hat 2 The Back" from the debut album Ooooooohhh… On the TLC Tip and settled in with "Kick Your Game" and "Girl Talk." There were many dance breaks throughout the night, especially during Left Eye's verses. The high energy delivered by the background dancers made it feel like a carefree house party. Lead by TLC, fans danced along to trending TikTok moves, which showed how in tune they are with their younger fans.
The group left the stage for a costume change and returned in black and silver shimmery outfits. They kept up the momentum as they performed "Ain't Too Proud To Beg" and "What About Your Friends" before pausing for a moment to speak to the young girls and women in the audience, to remind them that having "No Scrubs" is better than settling.
The energy of the crowd relaxed during the pair's next set as they began singing "American Gold," which was dedicated to frontline healthcare workers and U.S. soldiers — accompanied by videos and pictures on the screen. They kept it calm with "Unpretty," "Red Light Special," "Digging on You" and "Baby-Baby."
They ended the night with a dedication to Left Eye Lopes as they performed their other No. 1 single, "Waterfalls."
"We have to dedicate this one to our sister Lisa because this was her favorite song," T-Boz told the audience before taking in the sound of fans loudly and passionately rapping Left Eye's verses. After the song, Chilli and T-Boz walked gracefully across the stage talking to fans directly for several minutes, thanking them for showing up and for showing out.
Chilli chilled out with the crowd at the Pavilion at Toyota Music Factory on Tuesday.
KEEP THE DALLAS OBSERVER FREE... Since we started the Dallas Observer, it has been defined as the free, independent voice of Dallas, and we'd like to keep it that way. With local media under siege, it's more important than ever for us to rally support behind funding our local journalism. You can help by participating in our "I Support" program, allowing us to keep offering readers access to our incisive coverage of local news, food and culture with no paywalls.
David Archuleta Will Perform With 'Predominately Gay' Turtle Creek Chorale in Dallas
Photographer Curtis Smith Shot the Sex Pistols' Shitshow at the Longhorn Ballroom
'Running Away With Honor': Joni Mitchell's Hejira Remains Her True Masterpiece 45 Years Later
` }); } } ], "vars": { "placed": "12558143" } }); });
10 Dead Sara Songs You Must Hear Ahead of Their BFD Festival Appearance
By Vincent Arrieta
10 Best Concerts of the Week: Jonathan Tyler, Bell Biv DeVoe, Kansas and More
By David Fletcher
North Texas Electronic Artists and Labels To Look Out For in 2022
`) if (myParagraphTool.isBrParagraphBreakBeforeBlockElement(currentNode)) { myElement.find('> div').css({ width: '100%', minHeight:'30px', margin: '30px 0 30px 0', }); } else { myElement.find('> div').css({ width: '100%', minHeight:'30px', margin: '0 0 30px 0', }); } myParagraphTool.insertElemenAtIndex(myElement, i); } } }); apntag.anq.push(function() { if (typeof Foundation.ApnAds.fillAds === 'function') { Foundation.ApnAds.fillAds() } }) } } throwInExtraInsertions(12558143); jQuery(document).on('foundation:multiitem:itemLoaded', function (event) { throwInExtraInsertions(event.objId); }) } // if (typeof window.apntag === 'object' && typeof window.apntag.onEvent === 'function') { window.apntag.onEvent('adLoaded', reflowRightColumnItems) } // if (typeof twttr === 'object' && typeof twttr.ready === 'object' && typeof twttr.events === 'object') { twttr.events.bind('rendered', reflowRightColumnItems) } // jQuery(window).on('scroll', reflowRightColumnItems)


As someone who has handled research projects for AIRDA, I can visualize a crystal ball on timeshare. So in the future we could see smaller resorts moving out as a result of getting absorbed by the bigger players. I also think we will see new entrants, new products and changing service experiences.
There could be a consolidation of the organized sector in the timeshare industry, but the pie will expand, grow and diversify in my opinion.
And If we're wondering about growth, I think the key indicator here would be penetration of the timeshare industry. I'm referring to the proportion of the population using timeshare plans.
Ashok R Sankethi is an MR veteran who set up Kaybase in 2005. He has over 25 years of experience in the industry, and works on projects that call for higher levels of research and analytics. He is on the teaching faculty at leading business schools, and sits on the doctoral committees of several Ph. D scholars. Along with his colleague Poornima Bhaskaran, Ashok has published a book called Kay's Book of MR – which is the first book of case studies from an industry perspective. This book is now being used by faculty at over 15 business schools in the country.
Survey from PGi reveals that the workation is on the rise in America, with 82% of employees choosing to work on vacation in 2013. How do they connect? Mobile devices – smartphones leading the way, but more employees working on vacation with three devices – smartphones, tablets and laptops.
2009 was a tough one for the travel industry, but if the latest survey from Skyscanner is anything to go by, things are beginning to look up.
Almost half (49%) of the British travellers surveyed said they were planning to go abroad two or three times this year, but 16% said they intended to travel four or five times, while 13% said they would escape Blighty on six or more trips during 2010. 13% said they would travel once, while only 6% won't be going away at all.
LeisureLink, Inc., the leading distribution and revenue management platform for specialty lodging suppliers, today announced the first publication of the LeisureLink Demand Index™, showing dramatic growth in consumer demand for Vacation Lodging in mainstream online travel channels.
"Consumer demand in online channels has tripled over the last 2 years," said Steve Reich, Senior Vice president of LeisureLink. "Consumer acceptance of Vacation Lodging is growing rapidly, even with difficult economic conditions," he added. "This is a classic case of the 'network effect'— a hot product creates demand, demand creates distribution, and broad distribution creates more demand," said Reich. Vacation Lodging includes Vacation Rental Properties, Timeshare Resorts, and Boutique Hotels. "The growth in Vacation Lodging is in sharp contrast to the 7% decline in leisure and unmanaged business travel in 2009 recently estimated by research firm PhoCus Wright," he continued.
Proving that the holiday market is weathering the recession better than many other sectors, a company that organizes vacations for Americans 50 years or older has said the demand continues unabated.
Like many other industry players and commentators, Grand Circle says the economy is pushing people to seek exceptional travel values rather than cut back on vacations.
Approximately 2.3 million households state they are interested in purchasing some form of vacation time during the next two years.
Just two findings from 'Future Timeshare Buyers: 2009 Market Profile', which was developed from a nationally-representative sample of active leisure travelers who have expressed interest in acquiring some form of vacation time during the next two years, and supplemented through insights revealed in the Ypartnership/Yankelovich, Inc. 2009 National Leisure Travel MONITOR(SM). The study was authored exclusively for Interval International, which is releasing the results during the American Resort Development Association's (ARDA) 2009 Fall Conference, held in Washington, D.C.


Quite simply one of the best smartphones on the market, the HTC One A9 features a beautiful 5in edge-to-edge screen, fast Qualcomm Snapdragon octo-core processor, 16GB of memory all sitting on top of Android 6.0. And it takes snappy pics, too, thanks to a 13MP camera. You'd be paying over £300 for this handset in the shops!
The competition winner will be picked at random from all eligible entries that arrive by midnight of the closing date, December 22, 2015.
The winners will be notified via email by December 29, 2015. The winner must claim their prize within 7 days of the promoter sending notification. If the prize is unclaimed after this time, HEXUS reserves the right to offer the unclaimed prize to a substitute winner selected in accordance with these rules.
First prize is an HTC One A9 smartphone.
Doesn't look like it's on Facebook.
Anyway, my wife could use a new phone and this one looks good. I'll stick with a low end phone for now.
I have the 1st Gen M9, and trying to convince the other half to drop her Sony crud (which has been repaired more times than ANY of my HTC phones. 1 fault in 6 years.). This would be a great phone for her.
Thank you Hexus for all these cracking prizes.