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unitarily equivalent. In particular, { instead of}
\eqref{5.13} we have ${\sigma}_{ac}(A_{0})= {\sigma}_{ac}(\wt A)$.
\end{theorem}
\begin{proof}
Let $\wt {\Pi}_n = \{\cH_n, \wt{\Gamma}_{0n}, \wt{\Gamma}_{1n}\}$ be a
boundary triplet for ${ S^*}_n, \ n\in \N,$ defined according to
\eqref{3.8}, that is ${\wt\Gamma}_{0n}:= R_n\Gamma_{0n}$ and
${\wt\Gamma}_{1n}:=R^{-1}_n\bigl(\Gamma_{1n}- \re
(M_n(i))\Gamma_{0n}\bigr)$, where $R_n:=\sqrt{\im M_n(i))}.$ The
corresponding Weyl function ${\wt M}_n(\cdot)$ is
\begin{displaymath}
{\wt M}_n(z)=R^{-1}_n\bigl(M_n(z)- \re M_n(i)\bigr)R^{-1}_n, \quad
n\in \N.
\end{displaymath}
Since ${\wt M}_n(i)=i,\ n\in \N,$ by Theorem \ref{VI.3},
${\wt\Pi}=\oplus^{\infty}_{n=1}{\wt\Pi}_n=:\{\cH,{\wt\Gamma}_0,{\wt\Gamma}_1\}$
is a boundary triplet for $A^*=\oplus_{n=1}^{\infty} { S_n}^*$
satisfying
$A^*\upharpoonright{\xker\,}{\wt\Gamma}_0=A_0:=\oplus^{\infty}_{n=1}
{ S_{0n}}$. By definition of $\gotm^+_n(\cdot)$ and due to
Remark \ref{V.7} one has $\gotm^+_n(t) = \wt m^+_n(t) := \sup_{y
\in (0,1]}\|\wt M_n(t+iy)\|$ for $t \in \R$, $n \in \N$. Since
$A_0 =\oplus_{n=1}^{\infty} { S_{0n}}$ we get that $\wt
m^+(t) = \sup_n m^+_n(t)$, where $\wt m^+(t) := \sup_{y \in
(0,1]}\|\wt M(t + iy)\|$, $t \in \R$. Since, by assumption, the
maximal normal function $\wt m^+(t)$ is finite, we obtain from
Theorem \ref{V.5} that $\wt A^{ac}$ and $A^{ac}_0$ are unitarily
equivalent.
\end{proof}
\begin{corollary}\Label{VI.5}
Let the assumptions of Theorem \ref{VI.4} be satisfied and let
\begin{equation}\Label{6.12}
\cN := \{t \in \R:\ \sup_{n\in\N}\gotm^+_n(t) <
\infty\}.
\end{equation}
If condition \eqref{5.0} holds, then the parts $\wt{A}^{ac}E_{\wt
A}(\cN)$ and $A^{ac}_0E_{A_0}(\cN)$ of the operators
$\wt{A}$ and ${A_0},$ respectively, are unitarily equivalent.
\end{corollary}
Let $T$ and $T'$ be densely defined closed symmetric operators in
$\gotH$ and let $T_0$ and $T'_0$ be self-adjoint extensions of
$T$ and $T'$, respectively. It is said that the pairs $\{T,T_0\}$
and $\{T',T'_0\}$ are unitarily equivalent if there exists a
unitary operator $U$ in $\gotH$ such that $T' = UTU^{-1}$ and
$T'_0 = UT_0U^{-1}$.
\begin{corollary}\Label{VI.5a}
{ Assume the conditions of Theorem \ref{VI.4}. Let also} the
pairs $\{{ S_n},{ S_{0n}}\}$, $n\in \N,$ be unitarily
equivalent to the pair { $\{S_1, S_{01}\}$}. | -0.428278 | -1.574251 | -0.276709 | 1.846058 | [
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] | 1,332,863 | 37,888 | ArXiv | -1 | -1 |
If the maximal
normal function $m^+_1(t)$ is finite for a.e. $t \in \R$ and
condition \eqref{5.0} is satisfied, then the absolutely continuous
parts $\wt{A}^{ac}$ and $A^{ac}_0$ are unitarily equivalent.
\end{corollary}
\begin{proof}
Since the symmetric operators ${ S_n}$ are unitarily
equivalent, we assume without loss of generality that $\cH_n =
\cH$ for each $n \in \N$. Let $U_n$ be a unitary operator such
that $A_1 = U_n{ S_n}U^{-1}_n$ and $A_{01} = U_n{
S_{0n}}U^{-1}_n$. A straightforward computation shows that ${\Pi}'_n
:= \{\cH,{\Gamma}'_{0n},{\Gamma}'_{1n}\}$, ${\Gamma}'_{0n} := {\Gamma}_{01}U_n$ and
${\Gamma}'_{1n} := {\Gamma}_{1n}U_n$, defines a boundary triplet for ${
S^*_n}$. The Weyl function $M'_n(\cdot)$ corresponding to ${\Pi}'_n$
is $M'_n(z) = M_1(z)$. Hence $\gotm^+_n(\cdot)=
\gotm'^+_n(\cdot)$ and $\gotm^+_1(t) = \gotm'^+_n(t)$ for $t \in
\R,$ where $\gotm^+_n(t)$ and $\gotm'^+_n(t)$ are the invariant
maximal normal functions corresponding to the triplets ${\Pi}_n$
and ${\Pi}'_n$, respectively. By Remark \ref{V.7}(i), $\gotm^+_1(t)
= \gotm^+_n(t)$ for $t\in \R$ and $n \in \N$. Applying Theorem
\ref{VI.4} we complete the proof.
\end{proof}
\subsection{ Finite deficiency summands: $ac$-minimal extensions}
Here we improve the previous results assuming that
$n_{\pm}({ S_n})<\infty.$ First, we show that extensions
$A_0=\oplus^\infty_{n=1} { S_{0n}}(\in {\rm Ext\,}_A)$ of the form
\eqref{6.8} posses a certain spectral minimality property. To this
end we start with the following lemma.
\begin{lemma}\Label{V.9}
Let $H$ be a bounded non-negative self-adjoint operator in a
separable Hilbert space $\gotH$ and let $L$ be a bounded operator
in $\gotH$. Then
\item[\rm \;\;(i)] ${\rm dim}(\overline{{\rm ran\,}(H)}) =
{\rm dim}(\overline{{\rm ran\,}(\sqrt{H})})$.
\item[\rm \;\;(ii)]
If $L^*L \le H$, then
${\rm dim}(\overline{{\rm ran\,}(L)}) \le {\rm dim}(\overline{{\rm ran\,}(H)})$.
\item[\rm \;\;(iii)] If $P$ is an orthogonal projection,
then ${\rm dim}(\overline{{\rm ran\,}(PHP)}) \le
{\rm dim}(\overline{{\rm ran\,}(H)})$.
\end{lemma}
\begin{proof}
The assertion (i) is obvious.
(ii) If $L^*L \le H$, then there is a contraction $C$ such that
$L = C\sqrt{H}.$ Hence ${\rm dim}(\overline{{\rm ran\,}(L)}) =
{\rm dim}(\overline{{\rm ran\,}(C\sqrt{H})}) \le
{\rm dim}(\overline{{\rm ran\,}(\sqrt{H})}) = {\rm dim}(\overline{{\rm ran\,}(H)})$.
(iii) Clearly, ${\rm dim}(\overline{{\rm ran\,}(PHP)}) \le
{\rm dim}(\overline{{\rm ran\,}(HP)}) \le {\rm dim}(\overline{{\rm ran\,}(H)})$.
\end{proof}
\begin{theorem}\Label{VI.6}
Let $\{{ S_n}\}^\infty_{n=1}$ be a sequence of densely defined
closed symmetric operators, ${\rm dom\,}({ S_n})\subset \gotH_ | -0.060434 | -1.278798 | 0.245122 | 2.106195 | [
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] | 1,332,863 | 38,912 | ArXiv | -1 | -1 |
n$,
with $n_+({ S_n})=n_-({ S_n})< \infty$, $n \in \N$ and let
${ S_{0n}} = { S^*_{0n}}\in {\rm Ext\,}_{ S_n}$. Let also $A$
and $A_0$ be given by \eqref{6.5a} and \eqref{6.8}, respectively.
Then { $A_0$} is $ac$-minimal, in particular,
$\sigma_{ac}(A_0) \; { \subseteq} \;\sigma_{ac}(\wt A).$
\end{theorem}
\begin{proof}
By Theorem \ref{VI.3} there is a sequence of boundary triplets
${\Pi}_n := \{\cH_n,{\Gamma}_{0n},{\Gamma}_{1n}\}$, $n \in \N$, for ${
S^*_{n}}$ such that ${S_{0n}} = {
S^*_n}\upharpoonright{\xker\,}({\Gamma}_{0n})$, $n \in \N$, and the direct
sum ${\Pi} = \{\cH,{\Gamma}_0,{\Gamma}_1\} = \bigoplus^\infty_{n=1}{\Pi}_n$ of
the form \eqref{6.5a} is a boundary triplet for $A^*$ satisfying
$A_0 = A^*\upharpoonright{\xker\,}({\Gamma}_0)$. By Proposition
\ref{prop2.1}, any $\wt A = {\wt A}^*\in {\rm Ext\,}_A$ admits a
representation $\wt A = A_{\Theta}$ with ${\Theta} = {\Theta}^*\in \wt
\cC(\cH).$ By Corollary \ref{V.1A}(i), we can
{ assume that } $\wt A$ and
$A_0,$ { are disjoint, that is }
{ ${\Theta}= B =B^*\in \cC(\cH)$}. Consider the generalized Weyl
function $M_B(\cdot) := (B - M(\cdot))^{-1}$, where $M(\cdot) =
\bigoplus^\infty_{n=1}M_n(\cdot)$, cf. \eqref{6.7}. Clearly,
\begin{displaymath}
{\rm Im\,}(M_B(z)) = M_B(z)^*{\rm Im\,}(M(z))M_B(z), \quad z \in \C_+.
\end{displaymath}
Denote by $P_N$, $N \in \N$, the orthogonal projection from $\cH$
onto the subspace $\cH_N := \bigoplus^N_{n=1}\cH_n$. Setting
$M^{P_N}_B(z):= P_NM_B(z)\upharpoonright\cH_N$, { and taking
into account the block-diagonal form of $M(\cdot)$ and the
inequality ${\rm Im\,}(M(z))>0$ we obtain}
\begin{eqnarray}\Label{5.18}
\lefteqn{
{\rm Im\,}(M^{P_N}_B(z)) = {\rm Im\,}(P_NM_B(z)P_N) }\\
& & = P_NM_B(z)^*{\rm Im\,}(M(z))M_B(z)P_N \ge
M^{P_N}_B(z)^*{\rm Im\,}(M^{P_N}(z))M^{P_N}_B(z), \nonumber \end{eqnarray}
where $M^{P_N}(z) := P_NM(z)\upharpoonright\cH_N =
\bigoplus^N_{n=1}M_n(z)$. Since $P^N$ is a finite dimensional
projection the limits $M^{P_N}_B(t) := \,\mbox{\rm s-}\hspace{-2pt} \lim_{y\to+0}M^{P_N}_B(t+iy)$
and $M^{P_N}(t) := \,\mbox{\rm s-}\hspace{-2pt} \lim_{y\to+0}M^{P_N}(t+iy)$ exist for a.e. $t \in
\R$. From \eqref{5.18} we get
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] | 1,332,863 | 39,936 | ArXiv | -1 | -1 |
equation}\Label{5.19}
{\rm Im\,}(M^{P_N}_B(t)) \ge
M^{P_N}_B(t)^*{\rm Im\,}(M^{P_N}(t))M^{P_N}_B(t) \quad\text{ for
a.e.}\quad t \in \R.
\end{equation}
Since $M_B(\cdot)$ is { a} generalized Weyl function, it is a
strict $R_\cH$-function, that is, ${\xker\,}({\rm Im\,}(M_B(z)))=\{0\},\
z\in \C_+$. Therefore, $M^{P_N}_B(\cdot)$ is also strict. Hence
$0\in \varrho(M^{P_N}_B(z))$, $z\in \C_+$, and $G_N(\cdot) := -
(M^{P_N}_B(\cdot))^{-1}$ { is strict}. Since both $G_N(\cdot)$ and
$M^{P_N}_B(\cdot)$ are { matrix-valued} $R$-functions, the limits
$M^{P_N}_B(t+i0):= \lim_{y\to+0} M^{P_N}_B(t+iy)$ and $G_N(t+i0):=
\lim_{y\to+0} G_N(t+iy)$ exist for a.e. $t\in \R.$ Therefore,
passing to the limit in the identity $M^{P_N}_B(t+iy)G_N(t+iy) =
-I$ as $y\to 0,$ we get $M^{P_N}_B(t+i0)G_N(t+i0) = -I$ for a.e.
$t\in \R.$ Hence $M^{P_N}_B(t) := M^{P_N}_B(t+i0)$ is invertible
for a.e. $t \in \R$.
Further, combining \eqref{5.19} with Lemma \ref{V.9}(ii) we get
\begin{displaymath}
{\rm dim}\left(\overline{{\rm ran\,}\left(\sqrt{{\rm Im\,}
M^{P_N}(t)}M^{P_N}_B(t)\right)}\right) \le d_{M^{P_N}_B}(t) \quad
\text{for a.e.} \quad t \in \R.
\end{displaymath}
Since $M^{P_N}_B(t)$ is invertible for a.e. $t \in \R,$ we find
\begin{equation}\label{5.21}
d_{M^{P_N}}(t) := {\rm dim}\left(\overline{{\rm ran\,} \left( \sqrt
{{\rm Im\,} M^{P_N}(t)}\right)}\right) \le d_{M^{P_N}_B}(t)\quad
\text{for a.e.} \quad t \in \R.
\end{equation}
Let ${ D}_N = P_N \oplus { D}_0$ where { ${D}_0 \in
\gotS_2(\cH^\perp_N)$ and satisfy} ${\xker\,}({ D}_0) = {\xker\,}({D}^*_0) =
\{0\}$. Then ${\xker\,}({ D}_N) = {\xker\,}({D}^*_N) = \{0\}$ and $P_N =
P_N{D}_N = { D}_NP_N$. By Lemma \ref{V.9}(iii),
$d_{M^{P_N}}(t) \le d_{M^{{ D}_N}_B}(t)$ for a.e. $t \in \R$.
Further, for any ${D}\in \gotS_2(\cH)$ and satisfying ${\xker\,}({D}) =
{\xker\,}({D}^*) = \{0\},$ $d_{M_B^{D}}(t) = d_{M^{{D}_N}_B}(t)$ for
a.e. $t \in \R.$ { Combining this equality with \eqref{5.21}}
we get $d_{M^{P_N}}(t) \le d_{M_B^{ D}}(t)$ for a. | -0.432045 | -1.631551 | -0.312217 | 1.882477 | [
-0.21255402774882162,
-0.6515350027429243
] | [
-1.645876480072518,
-1.6172264227715627
] | [
-0.269734735597375,
-0.35469880994774805
] | [
2.002139726105781,
1.7628136614265153
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] | 1,332,863 | 40,960 | ArXiv | -1 | -1 |
e. $t \in \R$
and $N \in \N$. Since
\begin{equation}\Label{6.13A}
d_{M^{P_N}}(t) = \sum^N_{n=1}d_{M_n(t)} \quad \mbox{and} \quad
d_{M^D}(t) = \sum^\infty_{n=1}d_{M_n}(t)
\end{equation}
for a.e. $t \in \R,$ we finally prove that $d_{M^{ D}}(t) \le
d_{M_B^{D}}(t)$ for a.e. $t \in \R$. { One completes the
proof by applying Theorem} \ref{IV.10}(i).
\end{proof}
\begin{corollary}\Label{VI.7A}
Let { the assumptions} of Theorem \ref{VI.6} be satisfied and
{ let $S_n \ge 0$, $n \in \N$}. { Further, let $A$ and $A$
be given by \eqref{6.5a} and \eqref{6.8}, respectively}. Then the
Friedrichs and { the } Krein extensions $A^F$ and $A^K$ of
$A$ are $ac$-minimal. In particular, $(A^F)^{ac}$ and $(A^K)^{ac}$
are unitarily equivalent.
\end{corollary}
\begin{proof}
{ Combining Theorem \ref{VI.6} and Corollary \ref{cor5.5}
yields the assertion.}
\end{proof}
\begin{corollary}\Label{VI.7}
Let { the assumptions} of Theorem \ref{VI.6} be satisfied and let
\begin{equation}\Label{6.13}
\cD := \{t \in \R: \sum_{n\in\N}d_{M_n}(t) = \infty\}.
\end{equation}
If, { in addition,} condition \eqref{5.0} holds, then the parts $\wt A^{ac}E_{\wt
A}(\cD)$ and $A^{ac}_0E_{A_0}(\cD)$ of the operators $\wt A$ and
$A_0,$ respectively, are unitarily equivalent.
\end{corollary}
\begin{proof}
By \eqref{6.13A} and \eqref{6.13}, $d_{M^D}(t) = + \infty$ for
a.e. $t \in \cD$. Applying Theorem \ref{VI.6} and
Theorem \ref{II.4a}(ii) we complete the proof.
\end{proof}
\begin{corollary}\Label{VI.10}
{ Let the assumptions of Theorem \ref{VI.6} be satisfied and
let $\cN$ and $\cD$ be given by \eqref{6.12} and \eqref{6.13},
respectively. If condition \eqref{5.0} is valid, then} the {
parts} $\wt A^{ac}E_{\wt A}(\cD \cup \cN)$ and
$A^{ac}_0E_{A_0}(\cD\cup\cN)$ are unitarily equivalent.
\end{corollary}
\begin{proof}
By Corollary \ref{VI.5}, the parts $\wt A^{ac}E_{\wt A}(\cN)$ and
$A^{ac}_0E_{A_0}(\cN)$ are unitarily equivalent. Corollary
\ref{VI.7} yields the unitary equivalence of the parts
$\wt A^{ac}E_{\wt A}(\cD)$ and $A^{ac}_0E_{A_0}(\cD)$. Hence the
parts $\wt A^{ac}E_{\wt A}(\cD \cup \cN)$ and $A^{ac}_0E_{A_0}(\cD\cup\cN)$ are unitarily
equivalent { too}.
\end{proof}
\begin{corollary}\Label{VI.11}
{ Assume conditions of Theorem \ref{VI.6}}. { Then}
$\overline{\bigcup_{n\in\N}{\sigma}_{ac}({ S_{0n}})} \subseteq
{\sigma}_{ac}(\wt A).$ If, in addition, condition \eqref | -0.521416 | -1.617226 | -0.094734 | 1.814841 | [
-0.39129667123315987,
-0.6515350027429243
] | [
-1.5241137365434572,
-1.710339108999668
] | [
0.057440356677195714,
-0.2469085663689166
] | [
1.8668684721566307,
1.7628136614265153
] | 1,024 | [
512,
512
] | [
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278,
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319,
467,
29938,
960,
29892,
297,
6124,
29892,
4195,
320,
4915
] | 1,332,863 | 41,984 | ArXiv | -1 | -1 |
{5.0} is valid
and { the extensions $S_{0n}$, $n \in \N$, are purely
absolutely
continuous, then}
\begin{equation}\Label{6.17}
{\sigma}_{ac}(\wt A) = \overline{\bigcup_{n\in\N}{\sigma}_{ac}({ S_{0n}})}.
\end{equation}
\end{corollary}
\begin{proof}
The first statement { immediately follows} from Theorem \ref{VI.6}.
Relation \eqref{6.17} is implied by { Proposition} \ref{VI.5b}.
\end{proof}
\begin{corollary}\Label{VI.12}
{ Assume the conditions of Theorem \ref{VI.6}}. { Let
also the pairs} $\{{ S_n},{ S_{0n}}\}$, $n \in \N$, be
pairwise unitarily equivalent. { If condition \eqref{5.0}
holds }, then for any $\wt A\in {\rm Ext\,}_A$ the $ac$-parts $\wt
A^{ac}$ and $A^{ac}_0$
are unitarily equivalent.
\end{corollary}
\begin{remark}
{\rm
(i)\ For { the} special case { $n_\pm(S_n) = 1$, $n \in
\N$}, Theorem \ref{VI.6} complements \cite[Corollary 5.4]{ABMN05}
where the inclusion { $\sigma_{ac}(A_0)\subseteq
\sigma_{ac}(\wt A)$} was proved. Moreover, Corollary \ref{VI.12}
might be { regarded} as a substantial generalization of
\cite[Theorem 5.6(i)]{ABMN05} to the case $n_{\pm}({ S_n})>1$.
However, in the case $n_{\pm}({ S_n})=1,$ Corollary
\ref{VI.12} is implied by \cite[Theorem 5.6(i)]{ABMN05} where the
unitary equivalence of ${\wt A}^{ac} = {\wt A}_B^{ac}$ and
$A_0^{ac}$ was proved under { the weaker} assumption that
$B$ is purely singular. Indeed, by Proposition \ref{prop2.9}
condition \eqref{5.0} { with $\wt A = A_B$} is equivalent to
discreteness of $B.$
(ii)\ The inequality $N_{E^{ac}_{A_0}}(t) \le
N_{E^{ac}_{\wt{A}}}(t)$ in Theorem \ref{VI.6} might be strict even
for $t\in\sigma_{ac}(A_0)$. Indeed, assume that $(\alpha,\beta)$
is a gap for all { except for the operators}
$S_1,\ldots,S_N$. Set $S_1:=\oplus^N_{n=1} { S_n}$ and
$S_2:=\oplus^{\infty}_{ n=N+1}{ S_n}$. Then
$n_{\pm}(S_2)=\infty$ and $(\alpha,\beta)$ is a gap for $S_2.$ By
\cite{Bra04} there exists ${\wt S}_2={\wt S}^*_2\in {\rm Ext\,}_{S_2}$
having $ac$-spectrum within $(\alpha,\beta)$ of arbitrary
multiplicity. Moreover, even for operators $A = {
\oplus^\infty_{n=1} S_n}$ satisfying { assumptions} of
Corollary \ref{VI.12} with $n_{\pm}({ S_n})=1$ the inclusion
{ ${\sigma}_{ac}(A_0)\subseteq {\sigma}_{ac}(\wt A)$} might be strict
whenever condition \eqref{5.0} is violated, { cf. \cite{Bra04}
or \cite[{ Theorem 4.4}]{ABMN05} which guarantees the
appearance of prescribed spectrum either within one gap or within
several gaps of $A_0$}.
}
\end{remark}
\section{Sturm-Liouville operators with operator \\potentials}
\subsection{Bounded operator potentials}
Let $\cH$ be a separable Hilbert space. As usual, $L^2 | -0.32598 | -1.375492 | -0.012306 | 1.939707 | [
-0.47621654783108297,
-0.17574400059447992
] | [
-1.6029013941210848,
-1.1480817344684164
] | [
0.027005464372584484,
-0.051618007414327864
] | [
1.8044355857185614,
2.0749780936168616
] | 1,024 | [
512,
512
] | [
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] | 1,332,863 | 43,008 | ArXiv | -1 | -1 |
({\mathbb
R}_+,\cH) := L^2({\mathbb R}_+)\otimes\cH$ stands for the Hilbert
space of (weakly) measurable vector-functions $f(\cdot):{\mathbb
R}_+\to\cH$ satisfying $\int_{{\mathbb
R}_+}\|f(t)\|^2_{\cH}dt<\infty$. Denote also by $W^{2,2}({\mathbb
R}_+,\cH) := W^{2,2}({\mathbb R}_+)\otimes\cH$ the Sobolev space of
vector-functions taking values in $\cH$.
Let $T = T^* \ge 0$ be a bounded operator in $\cH$. Denote by
$A:=A_{\min}$ the minimal operator generated on $\gH:=L^2({\mathbb
R}_+,\cH)$ by a differential expression
$\cA=-\frac{d^2}{dx^2}\otimes I_{\cH}+I_{L^2({\mathbb R}_+)}\otimes
T$. It is known (see \cite{GG91,Rof-Bek69}) that $A$ is
given by
\begin{eqnarray}\Label{8.1}
(Af)(x) = -f''(x) + Tf(x), \quad f \in {\rm dom\,}(A) =
W^{2,2}_0({\mathbb R}_+,\cH),
\end{eqnarray}
where $W^{2,2}_0(\R_+,\cH) := \{f \in W^{2,2}(\R_+,\cH):\ f(0) =
f'(0) =0 \}$.
The operator $A$ is closed, symmetric and non-negative. It can be
proved similarly to \cite[Example 5.3]{BMN02} that $A$ is simple.
The adjoint operator $A^*$ is given by \cite[Theorem 3.4.1]{GG91}
\begin{equation}\Label{8.1B}
(A^*f)(x) = -f''(x) + Tf(x), \quad f \in {\rm dom\,}(A^*) =
W^{2,2}(\R_+,\cH).
\end{equation}
By \cite[Theorem 1.3.1]{LioMag72} the trace operators
$\Gamma_0, \ \Gamma_1: {\rm dom\,}(A^*)\to \cH,$
\begin{equation}\Label{8.2} {\Gamma}_0 f = f(0) \quad \mbox{and} \quad {\Gamma}_1 f = f'(0),
\quad f\in {\rm dom\,}(A^*), \end{equation}
are well defined Moreover, the deficiency subspace
$\mathfrak N_z(A)$ is
\begin{equation}\label{8.3A}
\mathfrak N_z(A) = \{e^{ix\sqrt{z-T}}h: \ h \in \cH\}, \qquad z\in
\C_{\pm}.
\end{equation}
\begin{lemma}\label{lem6.1}
A triplet ${\Pi} = \{\cH,{\Gamma}_0,{\Gamma}_1\}$, with ${\Gamma}_0$ and ${\Gamma}_1$
defined by \eqref{8.2}, forms
a boundary triplet for $A^*$.
The corresponding Weyl function $M(\cdot)$ is
\begin{equation}\Label{8.2B} M(z) = i\sqrt{z-T} = i\int\sqrt{t+iy -
{\lambda}}\;dE_T({\lambda}), \quad z = t+iy \in \C_+.
\end{equation}
\end{lemma}
\begin{proof}
One obtains the Green formula integrating by parts. The
surjectivity of the mapping ${\Gamma}:=(\Gamma_0,\Gamma_1)^\top:
{\rm dom\,}(A^*) \rightarrow \cH \oplus \cH$ is immediate from
\eqref{8.2} and \cite[Theorem 1.3.2]{LioMag72}. Formula
\eqref{8.2B} is implied by \eqref{8.3A}.
\end{proof}
\begin{lemma}\Label{VII.1}
Let $T$ be a bounded non-negative self-adjoint operator | 0.304498 | -1.072875 | 0.278093 | 2.616063 | [
0.38393962158013856,
0.22505727181628235
] | [
-1.1803130489319913,
-0.9654376191748251
] | [
0.35925303869792374,
0.19693361307333052
] | [
2.824172730873692,
2.4079534879532307
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] | 1,332,863 | 44,032 | ArXiv | -1 | -1 |
in $\cH$
and let $A$ and $\Pi=\{\cH,\Gamma_0,\Gamma_1\}$ be defined by
\eqref{8.1} and \eqref{8.2}, respectively. Then
\item[\rm\;\;(i)] the invariant maximal normal function $\gotm^+(t)$
of the Weyl function $M(\cdot)$ is finite for all $t \in \R$ and
satisfies
\begin{equation}\Label{7.3a}
\gotm^+(t) \le 2(1 + t^2)^{1/4}, \quad t \in \R.
\end{equation}
\item[\rm\;\;(ii)] The limit $M(t+i0) := \,\mbox{\rm s-}\hspace{-2pt} \lim_{y\to+0}M(t+iy)$
exists, is bounded and equals
\begin{equation}\Label{7.3}
M(t +i0) = i\int_\R \sqrt{t-{\lambda}} dE_T({\lambda}) \quad \text{for any}\ \
t \in \R.
\end{equation}
\item[\rm\;\;(iii)] $d_M(t) = {\rm dim}({\rm ran\,}(E_T([0,t))))$ for any $t
\in \R$.
\end{lemma}
\begin{proof} (i)
It is immediate from \eqref{8.2B} and definition \eqref{4.12A} of
$\gotm^+(\cdot)$ that
\begin{displaymath} \gotm^+(t) \le \sup_{y\in(0,1]}\sup_{{\lambda} \ge 0}
\left|\frac{\sqrt{t + iy -{\lambda}} -
{\rm Re\,}(\sqrt{i-{\lambda}})}{{\rm Im\,}(\sqrt{i - {\lambda}})}\right|. \end{displaymath}
Clearly, $\sqrt{i-{\lambda}} = (1 + {\lambda}^2)^{1/4}e^{i(\pi -\varphi)/2}$
where $ \varphi := \arccos\left(\tfrac{{\lambda}}{\sqrt{1 +
{\lambda}^2}}\right).$ Hence
\begin{displaymath} \left|\frac{\re(\sqrt{i-{\lambda}})}{\im(\sqrt{i-{\lambda}})}\right| =
\tan(\frac{\varphi}{2}) = \frac{1}{{\lambda} + \sqrt{1 + {\lambda}^2}} \le 1,
\quad {\lambda} \ge 0. \end{displaymath}
Furthermore, we have
\begin{displaymath} \left|\frac{\sqrt{t + iy -{\lambda}}}{\im(\sqrt{i - {\lambda}})}\right|
\le \sqrt{2}\sqrt{\frac{\sqrt{({\lambda} - t)^2 + y^2}}{{\lambda} + \sqrt{1 +
{\lambda}^2}}} \le 2^{3/4}(1 + t^2)^{1/4} \end{displaymath}
for ${\lambda} \ge 0$, $t \in \R$ and $y \in (0,1]$ which yields
\eqref{7.3a}.
(ii) From \eqref{8.2B} we find $M(t):= M(t +i0) := \,\mbox{\rm s-}\hspace{-2pt} \lim_{y\to+0}
i\sqrt{t+iy - T} = i\sqrt{t-T},$ for any $t \in \R,$ which proves
\eqref{7.3}. Clearly, $M(t)\in [\cH]$ since $T\in [\cH].$
(iii) It follows that $\im(M(t)) = \sqrt{t-T}E_T([0,t))$, which
yields $d_M(t) = {\rm dim}({\rm ran\,}(\im(M(t))))= {\rm dim}({\rm ran\,}(E_T([0,t))))$.
\end{proof}
With the operator $A=A_{\min}$ it is naturally associated a
(closable) quadratic form ${\mathfrak t}'_F[f]:= (Af,f),\
{\rm dom\,}({\mathfrak t}')={\rm dom\,}(A).$ Its closure ${\mathfrak | -0.065432 | -1.264473 | 0.056172 | 1.970923 | [
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] | 1,332,863 | 45,056 | ArXiv | -1 | -1 |
t}_F$ is
given by
\begin{equation}\Label{8.1A} { \mathfrak t_F[f]} := \int_{\R_+}
\left\{\|f'(x)\|^2_\cH + \|\sqrt{T}f(x)\|^2_\cH\right\}dx, \end{equation}
$f \in {\rm dom\,}({\mathfrak t}_F) = W^{1,2}_0(\R_+,\cH)$,
where $W^{1,2}_0(\R_+,\cH) := \{W^{1,2}(\R_+,\cH): f(0)=0\}.$
By definition, the Friedrichs extension $A^F$ of $A$ is a
self-adjoint operator associated with ${\mathfrak t}_F$. Clearly,
$A^F = A^*\upharpoonright ({\rm dom\,}(A^*)\cap{\rm dom\,}({\mathfrak t}_F)).$
\begin{theorem}\label{prop7.2A}
Let $T\ge 0$, $T = T^*\in [\cH]$, and $t_0:=\inf{\sigma}(T)$.
Let $A$ be defined by \eqref{8.1} and ${\Pi} = \{\cH,{\Gamma}_0,{\Gamma}_1\}$
the boundary triplet for $A^*$ defined by \eqref{8.2}.
Then
\item[\rm \;\;(i)] { the} Friedrichs extension $A^F$ coincides
with $A_0$ { that is}
\begin{displaymath} {\rm dom\,}(A^F) = {\rm dom\,}(A^*)\cap{\rm dom\,}({\mathfrak t}_F)= \{f \in
W^{2,2}(\R_+,\cH):\ f(0) = 0\}= {\rm dom\,}(A_0), \end{displaymath}
{ and } $A^F$ corresponds to the Dirichlet problem. Moreover,
$A^F$ is absolutely continuous, $A^F = (A^F)^{ac}$, and
${\sigma}(A^F)={\sigma}_{ac}(A^F)= [t_0,\infty).$
\item[\rm \;\;(ii)] { the} Krein extension $A^K$ is given by
\begin{equation}\label{7.4C}
{\rm dom\,}(A^K)=\{f\in W^{2,2}({\mathbb R}_+,\cH):\ f'(0) +
\sqrt{T}f(0)=0\}.
\end{equation}
Moreover, ${\xker\,}(A^K) = \gH_0 := \overline{\gH'_0}$, $\gH'_0:=
\{e^{-x\sqrt{T}}h:\ h\in {\rm ran\,}(T^{1/4})\}$ and the restriction
$A^K\upharpoonright{\rm dom\,}(A^K) \cap \gH^{\perp}_0$ is absolutely
continuous, that is $\gH^{\perp}_0=\gH^{ac}(A^{K})$ and $A^K=
{0}_{\gH_0}\oplus (A^K)^{ac}$.
\item[\rm\;\;(iii)] The extension $A_1 := A^*\upharpoonright
{\xker\,}({\Gamma}_1)$, { coincides with $A^N$}, ${\rm dom\,}(A^N) := \{f \in
W^{2,2}(\R_+,\cH): f'(0) = 0\}$, { i.e. $A_1$ corresponds to
the Neumann boundary condition}. Moreover, $A^N$ is absolutely
continuous $(A^N)^{ac} = A^N$ and ${\sigma}(A^N) = {\sigma}_{ac}(A^N) =
[t_0,\infty)$.
\item[\rm \;\;(iv)] The operators $A^F$, $(A^K)^{ac}$ and $A^N$ are
unitarily equivalent.
\end{theorem}
\begin{proof}
(i) Let ${\Pi} = \{\cH,{\Gamma}_0,{\Gamma}_1\}$ be the boundary triplet
defined { in } Lemma \ref{lem6.1}. We show that $A^F = A_0
:= A^*\upharpoonright{\ | 0.080557 | -0.913509 | 0.465775 | 2.397548 | [
0.1634218775113381,
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] | 1,332,863 | 46,080 | ArXiv | -1 | -1 |
xker\,}({\Gamma}_0)$. It follows from \eqref{8.1B}
and \eqref{8.2} that ${\rm dom\,}(A_0) = \{f \in W^{2,2}(\R_+,\cH): f(0)
= 0\}$. Since ${\rm dom\,}(A_0)\subset W^{1,2}_0(\R_+,\cH)=
{\rm dom\,}({\mathfrak t}_F),$ we have $A_0 = A^F$ (see \cite[Section
8]{AG81} and \cite[Theorem 6.2.11]{Ka76}).
It follows from \eqref{7.3} and \cite[Theorem 4.3]{BMN02} that
${\sigma}_p(A_0) = {\sigma}_{sc}(A_0) = \emptyset$. Hence $A_0$ is
absolutely continuous. Taking into account Lemma \ref{VII.1}(iii)
and Proposition \ref{III.8} we get ${\sigma}(A_0) = {\sigma}_{ac}(A_0) =
{\rm cl}_{ac}({\rm supp\,}(d_M)) = [t_0,\infty)$ which proves (i).
(ii)\ By \cite[Proposition 5]{DM91} $A^K$ is defined by $A^K=
A^*\upharpoonright {\xker\,}({\Gamma}_1 - M(0){\Gamma}_0).$ It follows from \eqref{8.2B}
that $M(0)=-\sqrt{T}$. Therefore, $A^K$ is defined by \eqref{7.4C}.
It follows from the extremal property of the Krein extension that
${\xker\,}(A^K) ={\xker\,}(A^*)$. Clearly, $f_h(x) := \exp(-x\sqrt{T})h\in
L^2(\R_+,\cH),$ $h\in {\rm ran\,}(T^{1/4}),$ since
\begin{equation*}
\int^{\infty}_0\!\|\exp(-x\sqrt{T})h\|^2_{\cH}dx=
\int^{\|T\|}_0\!d\rho_h(t)\int^{\infty}_0
\!e^{-2x\sqrt{t}}dx=\int^{\|T\|}_0 \!\frac{1}{2\sqrt
t}d\rho_h(t)<\infty,
\end{equation*}
where $\rho_h(t):=\bigl(E_T(t)h,h\bigr)$. Thus,
$\gH'_0\subset {\xker\,}(A^*).$ It is easily seen that $\gH'_0$ is
dense in $\gH_0.$ To investigate the rest of the spectrum of
$A^K$ consider the Weyl function $M_K(\cdot)$ corresponding to
$A^K$. It follows from \eqref{8.2B} and Proposition \ref{prop3.1}
that
\begin{eqnarray*}
\lefteqn{
M_K(z)=M_{-\sqrt{T}}(z)=-\bigl(\sqrt{T}+M(z)\bigr)^{-1}}\\
& &
= -(\sqrt{T}+i\sqrt{z-T})^{-1}
=\frac{1}{z}(i\sqrt{z-T}-\sqrt{T}) = -\frac{2\sqrt{T}}{z}+\Phi(z).
\nonumber
\end{eqnarray*}
where $\Phi(z):=\frac{1}{z}[i\sqrt{z-T}+\sqrt{T}]$.
It follows that for $t>0$
\begin{equation}\label{7.13}
\im M_K(t+i0) = \im\Phi(t+i0)=t^{-1}\sqrt{t-T}E_T([0,t)).
\end{equation}
Hence, by \cite[Theorem4.3]{BMN02}, $\sigma_p(A^K)\cap(0,\infty)=
\sigma_{sc}(A^K)\cap(0,\infty)=\emptyset$. It follows from
\eqref{7.13} that $\im{(M_K(t+i0))}>0$ for $t>t_0$. By Proposition
\ref{III. | -0.076785 | -1.420257 | -0.254517 | 2.168627 | [
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] | 1,332,863 | 47,104 | ArXiv | -1 | -1 |
8} $\sigma_{ac}(A^K)= [t_0,\infty)$. Further, it follows
from \eqref{7.3} and \eqref{7.13} that for any $t>t_0$
\begin{eqnarray*}
\lefteqn{
d_M(t)= {\rm rank\,}(\im(M(t)))} \\
& & = {\rm rank\,}(E_T([0,t))) = {\rm rank\,}(\im(M_K(t)))= d_{M_K}(t)
\end{eqnarray*}
Combining this equality with $\sigma_{ac}(A^K)= \sigma_{ac}(A^F)=
[t_0,\infty),$ we conclude that $A^F$ and $(A^K)^{ac}$ are unitarily
equivalent.
(iii) By Proposition \ref{prop3.1} the Weyl function corresponding to
$A_1= A^*\upharpoonright {\xker\,}({\Gamma}_1 - 0{\Gamma}_0)$ is
\begin{displaymath}
M_0(z) := (0 -M(z))^{-1} = i(z - T)^{-1/2} = i\int
\frac{1}{\sqrt{z -{\lambda}}}dE_T({\lambda}), \quad z \in \C_+.
\end{displaymath}
Since $M_0(\cdot)$ is regular within $(-\infty, t_0),$ we have
$(-\infty, t_0)\subset \varrho(A_1)$. Further, let $\tau
> t_0$. We set $\cH_\tau := E_T([t_0,\tau))\cH$ and note that for any $h \in
\cH_\tau$ and $t > \tau$
\begin{equation}\label{7.14}
\bigl(M_0(t+i0)h,h\bigr) = i\bigl((t-T)^{-1/2}h,h\bigr)=
i\int_{t_0}^{\tau} \frac{1}{\sqrt{t -{\lambda}}}d(E_T({\lambda})h,h).
\end{equation}
Hence for any $h\in \cH_\tau\setminus\{0\}$ and $t > \tau$
\begin{displaymath}
0 <
(t-t_0)^{-1/2}\|h\|^2 \le {\rm Im\,}(M_0(t+i0)h,h) = \int_{t_0}^{\tau}
(t -{\lambda})^{-1/2}d(E_T({\lambda})h,h) <\infty.
\end{displaymath}
By \cite[Proposition 4.2]{BMN02}, ${\sigma}_{ac}(A_1) \supseteq
[\tau,\infty)$ for any $\tau > t_0,$ which yields ${\sigma}_{ac}(A_1) =
[t_0,\infty)$. It remains to show that $A_1$ is purely
absolutely continuous. Since $M_0(t+i0)\not\in [\cH]$ we cannot
apply \cite[Theorem 4.3]{BMN02} directly. Fortunately, to
investigate the $ac$-spectrum of $A_1$ we can use \cite[Corollary
4.7]{BMN02}. For any $t \in \R,$ \ $y > 0,$ and $h \in \cH$ we
set
\begin{displaymath} V_h(t+iy) := \im(M_0(t+iy)h,h) =
\int\im\left(\frac{1}{\sqrt{{\lambda}-t-iy}}\right)d(E_T({\lambda})h,h). \end{displaymath}
Obviously, one has
\begin{displaymath} V_h(t+iy) \le \int\frac{1}{(({\lambda}-t)^2 +
y^2)^{1/4}}d(E_T({\lambda})h,h), \quad t \in \R, \quad y > 0, \quad h
\in \cH. \end{displaymath}
Hence
\begin{displaymath} V_h(t+iy)^p \le \|h\|^{2(p-1)} \ | -0.087314 | -1.34147 | -0.098538 | 2.080181 | [
-0.11958730800553077,
-0.055041353413964184
] | [
-1.3020757924610522,
-1.3808634500386798
] | [
-0.0896616227950919,
-0.10741530997278179
] | [
2.022950688251804,
2.1374109800549306
] | 1,024 | [
512,
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] | 1,332,863 | 48,128 | ArXiv | -1 | -1 |
int\frac{1}{(({\lambda}-t)^2 +
y^2)^{p/4}}d(E_T({\lambda})h,h), \quad p \in (1,\infty). \end{displaymath}
We show that for $p \in (1,2)$ and $-\infty < a < b < \infty$
\begin{displaymath} C_p(h; a,b) := \sup_{y\in (0,1]}\int^b_a V_h(t+iy)^p\;dt <
\infty. \end{displaymath}
Clearly,
\begin{eqnarray*}
\int^b_a V_h(t+iy)^p dt \le \|h\|^{2(p-1)}\;\int^{\|T\|}_0
d(E({\lambda})h,h) \int^b_a \frac{1}{(({\lambda}-t)^2 + y^2)^{p/4}}dt \nonumber \\
= \|h\|^{2(p-1)}\;\int^{\|T\|}_0 d(E({\lambda})h,h) \int^{b-{\lambda}}_{a-{\lambda}}
\frac{1}{(t^2 + y^2)^{p/4}}dt. \quad
\end{eqnarray*}
Note, that for $p \in (1,2)$ and $-\infty < a < b < \infty$
\begin{equation*}
\int^{b-{\lambda}}_{a-{\lambda}} \frac{1}{(t^2 + y^2)^{p/4}}dt \le
\int^{b}_{a-\|T\|} \frac{1}{t^{p/2}}dt =: \varkappa_p(b,a-\|T\|) <
\infty,
\end{equation*}
Hence $C_p(h;a,b) \le \varkappa_p(b,a-\|T\|)\|h\|^{2p} < \infty$
for $p \in (1,2)$, $-\infty < a < b < \infty$ and $h \in \cH$. By
\cite[Corollary 4.7]{BMN02}, $A_1$ is purely absolutely
continuous on any bounded interval $(a,b)$. Hence $A_1$ is purely
absolutely continuous.
(iv) It follows from \eqref{7.3} and \eqref{7.13} that
$d_M(t)=d_{M_K}(t)= {\rm rank\,}(\sqrt{t-T})$ {for} $t>t_0$.
Combining this equality with $\sigma_{ac}(A^K)= \sigma_{ac}(A^F)=
[t_0,\infty),$ we conclude that $A^F$ and $(A^K)^{ac}$ are unitarily
equivalent.
Passing to $A_1,$ we assume that $1 \le {\rm dim}({\rm ran\,}(E_T([0,s))))=p_1
< \infty$ for some $s > 0$. Let ${\lambda}_k$, $k \in \{1,\ldots,p\}, \
p\le p_1$, be the set of distinct eigenvalues within $[0,s)$.
Since $M_0(t+iy)E_T([0,t))$ is the $p\times p$ matrix-function,
the limit $M_0(t+i0)E_T([0,t))$ exists for $t \in [0,s) \setminus
\bigcup^p_{k=1}\{{\lambda}_k\}.$ It follows from \eqref{7.14} that
\begin{displaymath}
\im(M_0(t)) = |T-t|^{-1/2}E_T([0,t)),\qquad t \in [0,s)
\setminus \bigcup^p_{k=1}\{{\lambda}_k\}.
\end{displaymath}
This yields
\begin{displaymath}
d_{M_0(t))} := {\rm dim}({\rm ran\,}(\im(M_0(t)))) =
{\rm dim}({\rm ran\,}(E_T([0,t)))) = d_M(t)
\end{displaymath}
for a.e $t \in [0,s) \setminus \bigcup^p_{k=1}\{{\lambda}_k | -0.320074 | -1.244776 | -0.035132 | 1.929301 | [
-0.43512628496112016,
-0.20502081288932844
] | [
-1.4524885932910685,
-1.0370627624272137
] | [
-0.33821324328275026,
0.26794836178409004
] | [
1.9397068396677115,
1.9188958775216884
] | 1,024 | [
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] | 1,332,863 | 49,152 | ArXiv | -1 | -1 |
\}$, that
is, for a.e. $t \in [0,s)$.
If ${\rm dim}(E_T([t_0,s))) = \infty$, then there exists a point
$s_0\in (0, s),$ such that ${\rm dim}(E_T([0,s_0])) = \infty$ and
${\rm dim}(E_T([0,s))) < \infty$ for $s \in [0,s_0)$. For any $t \in
(s_0,s)$ choose $\tau \in (s_0,t)$ and note that
${\rm dim}({\rm ran\,}(E_T([0,\tau)))) = \infty$. We set $\cH_\tau :=
E_T([0,\tau))\cH$ and $\cH_\infty := E_T([\tau,\infty))\cH$ as
well as $T_\tau := TE_T([0,\tau))$ and $T_\infty :=
TE_T([\tau,\infty))$. Further, we choose Hilbert-Schmidt
operators $D_\tau$ and ${D}_\infty$ in $\cH_\tau$ and
$\cH_\infty$, respectively, such that ${\xker\,}(D_\tau) =
{\xker\,}(D^*_\tau) = {\xker\,}(D_\infty) = {\xker\,}(
D^*_\infty) = \{0\}$. According to the decomposition
$\cH=\cH_\tau \oplus\cH_\infty$ we have $M_0 = M_\tau
\oplus M_\infty,$ ${ D} = { D}_\tau \oplus{ D} _\infty$ and
$d_{M_0^{ D}}(t) = d_{M_\tau^{{ D}_\tau}}(t) + d_{M_\infty^{
D_\infty}}(t)$ for a.e. $t \in [0,\infty).$
Hence $d_{M_0^{D}}(t) \ge d_{M_\tau^{{ D}_\tau}}(t)$ for a.e. $t
\in [0,\infty)$. Clearly, $M_\tau (t+iy) = i(t + iy -
T_\tau)^{-1/2}.$ If $t > \tau$, then $t\in \varrho(T_\tau)$ and
$M(t) := \,\mbox{\rm s-}\hspace{-2pt} \lim_{y\to+0} M(t+i0)$ exists and
\begin{displaymath}
M_\tau(t) := \,\mbox{\rm s-}\hspace{-2pt} \lim_{y\to 0}M_\tau(t+iy) = i(t -
T_\tau)^{-1/2}E_T([0,\tau)).
\end{displaymath}
Hence $d_{M_\tau^{D_\tau}}(t) = {\rm dim}({\rm ran\,}(E_T([0,\tau)))) =
\infty$ for $t > s_0$. Hence $d_{M_0^D}(t) = d_M(t) = \infty$ for a.e.
$t > s_0$ which yields $d_{M_0^D}(t) = d_M(t)$ for a.e. $t \in
[0,\infty)$. Using Theorem \ref{IV.10}(ii) we obtain that $A^{ac}_0$
and $A^{ac}_1$ are unitarily equivalent which shows $A_0$ and $A_1$
are unitarily equivalent.
\end{proof}
Next we describe the spectral properties of any self-adjoint
extension of $A.$ In particular, we show that the
Friedrichs extension $A^F$ of $A$ is $ac$-minimal, though $A$ does
not satisfy conditions of Theorem \ref{VI.6}.
\begin{theorem}\Label{VII.3}
Let $T\ge 0$, {$T = T^*\in [\cH]$}, and $t_1:=\inf{\sigma}_{{\rm ess\,}}(T)$.
Let also $A$ be the symmetric operator defined by \eqref{8.1} and
$\wt A = \wt A^* \in {\rm Ext\,}_A$. Then
\item[\rm\;\;(i)] the absolutely continuous part $\wt A^{ac}
E_{\wt A}([t_1,\infty))$ of $\wt | -0.406363 | -1.577833 | -0.150531 | 1.840855 | [
-0.5008707055530607,
-0.3118554963512317
] | [
-1.4954636792425018,
-1.6602015087229958
] | [
-0.15560388945508288,
-0.14545892535354582
] | [
1.700380774988446,
1.9813287639597577
] | 1,024 | [
512,
512
] | [
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310,
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] | 1,332,863 | 50,176 | ArXiv | -1 | -1 |
A E_{\wt A}([t_1,\infty))$ is
unitarily equivalent to $A^FE_{A^F}([t_1,\infty)) =
(A^F)^{ac}E_{A^F}([t_1,\infty))$;
\item[\rm\;\;(ii)] the Friedrichs extension $A^F$ is
$ac$-minimal { and} $\sigma_{ac}(A^F)\subseteq
\sigma_{ac}(\wt A)$;
\item[\rm \;\;(iii)] the absolutely continuous part $\wt A^{ac}$
of $\wt A$ is unitarily equivalent to $A^F$ whenever either $(\wt
A - i)^{-1} - (A^F - i)^{-1} \in \gotS_\infty(\gotH)$ or $(\wt A -
i)^{-1} - (A^K - i)^{-1} \in \gotS_\infty(\gotH)$.
\end{theorem}
\begin{proof}
By Corollary \ref{V.1A} it suffices to assume that the
extension $\wt A = \wt A^*$ is disjoint with $A_0,$ that is, by
Proposition \ref{prop2.1}(ii) it admits a representation $\wt A
= A_B$ with $B\in \cC(\cH).$
(i) Let $\Pi=\{\cH, {\Gamma}_0,{\Gamma}_1\}$ be a boundary triplet for $A^*$
defined by \eqref{8.2}. In accordance with Theorem \ref{IV.10}
we calculate $d_{M_B^K}(t)$ where $M_B(\cdot) :=
(B-M(\cdot))^{-1}$ is the generalized Weyl function of
the extension $A_B$. Clearly,
\begin{equation}\label{7.15}
\im(M_B(z)) = M_B(z)^*\im(M(z))M_B(z), \quad z \in
\C_+. \end{equation}
Since $\re(\sqrt{z - {\lambda}})>0$ for $z = t +iy,\ y>0,$ it
follows from \eqref{8.2B} that
\begin{equation}\label{7.16}
\im(M(z)) = \int_{[0,\infty)}\re(\sqrt{z -
{\lambda}})\;dE_T({\lambda}) \ge \int_{[0,\tau)}\re(\sqrt{z -
{\lambda}})\;dE_T({\lambda}),
\end{equation}
where $z = t +iy$. It is easily seen that
\begin{equation} \label{7.16A}
\re(\sqrt{z - {\lambda}}) \ge \sqrt{t-{\lambda}} \ge
\sqrt{t-\tau}, \quad {\lambda} \in [0,\tau), \quad t > \tau.
\end{equation}
Combining \eqref{7.15} with \eqref{7.16} and \eqref{7.16A} we get
\begin{displaymath} \im(M_B(t+iy)) \ge
\sqrt{t-\tau}M_B(t+iy)^*E_T([0,\tau))M_B(t+iy), \quad t > \tau
>0. \end{displaymath}
Let $Q$ be a finite-dimensional orthogonal projection, $Q \le
E_T([0,\tau))$. Hence
\begin{displaymath} \im(M_B(t+iy)) \ge \sqrt{t-\tau}M_B(t+iy)^*QM_B(t+iy), \quad
t > \tau >0, \quad y > 0. \end{displaymath}
Setting $\cH_1= {\rm ran\,}(Q)$, $\cH_2 := {\rm ran\,}(Q^\perp)$, and
choosing $K_2\in \mathfrak S_2(\cH_2)$ and satisfying ${\xker\,}(K_2)
= {\xker\,}(K^*_2) = \{0\}$, we define a Hilbert-Schmidt operator $K :=
Q \oplus K_2\in \mathfrak S_2(\cH).$ Clearly, ${\xker\,}(K) =
{\ | -0.268625 | -1.534858 | -0.192379 | 2.064573 | [
-0.3584244609371896,
-0.17882577030972713
] | [
-1.4453260789658295,
-1.6243889370968014
] | [
-0.1479951663789301,
-0.2367636022673795
] | [
2.199843866493,
1.9293013585947
] | 1,024 | [
512,
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] | 1,332,863 | 51,200 | ArXiv | -1 | -1 |
xker\,}(K^*) = \{0\}$ and,
\begin{eqnarray}\label{6.14A}
\lefteqn{
\im(K^*M_B(t+iy)K) \ge }\\
& &
\sqrt{t-\tau}K^*M_B(t+iy)^*QM_B(t+iy)K, \quad t > \tau >0.
\nonumber
\end{eqnarray}
Since $M_B(\cdot)\in (R_\cH)$ and $Q,\ K\in \mathfrak
S_2(\cH),$ the limits
\begin{eqnarray*}
K^*M_B(t)^*Q & := & \,\mbox{\rm s-}\hspace{-2pt} \lim_{y\to+0}K^*M_B(t + iy)^*Q \quad \mbox{and} \quad \\
(QM_BK)(t) & := & \,\mbox{\rm s-}\hspace{-2pt} \lim_{y\to+0}QM_B(t+iy)K
\end{eqnarray*}
exist for a.e. $t\in \R$ (see \cite{BirEnt67}). Therefore
passing to the limit as $y\to 0$ in \eqref{6.14A}, we arrive at
the inequality
\begin{displaymath}
\im(M_B^K(t)) \ge \sqrt{t-\tau}(K^*M_B(t)^*Q)(QM_BK(t)), \quad t
> \tau >0, \quad y> 0.
\end{displaymath}
It follows that
\begin{equation}\label{7.18}
{\rm dim}({\rm ran\,}\left((QM_BK)(t)\right)) \le
{\rm dim}({\rm ran\,}\left(\im M_B^K(t)\right)) =d_{M^K_B}(t), \quad t > \tau.
\end{equation}
We set $\wt M^Q_B(z) :=QM_B(z)Q \upharpoonright\cH_1.$
Since ${\rm dim}(\cH_1)<\infty$ the limit $\wt M^Q_B(t):=
\,\mbox{\rm s-}\hspace{-2pt} \lim_{y\to+0}\wt M^Q_B(t+iy)$ exists for a.e. $t\in \R.$
Since $(QM_BK)(t)\upharpoonright\cH_1 = {\rm ran\,}\left((\wt
M^Q_B)(t)\right)$, \eqref{7.18} yields the inequality
\begin{equation}\Label{7.18a}
{\rm dim}({\rm ran\,}\left(\wt M^Q_B(t)\right)) \le
{\rm dim}({\rm ran\,}\left((QM_BK)(t)\right)) \le d_{M^K_B}(t)
\end{equation}
for a.e. $t \in [\tau, \infty)$.
Since ${\rm dim}(\cH_1) <\infty$ and ${\xker\,}(\wt M^Q_B(z)) =\{0\},
z\in \C,$ we easily get by repeating the corresponding reasonings
of the proof of Theorem \ref{VI.6} that ${\rm ran\,}\left(\wt
M^Q_B(t)\right) = \cH_1$ for a.e. $t \in \R$. Therefore
\eqref{7.18a} yields ${\rm dim}(\cH_1) \le d_{M^K_B}(t)$ for a.e. $t
\in [\tau, \infty).$
If $\tau > t_1$, then ${\rm dim}(E_T([0,\tau))\cH) =\infty$ and
the dimension of a projection $Q \le E_T([0,\tau))$ can be
arbitrary. Thus, $d_{M_B^K}(t) = \infty$ for a.e. $t >
\tau$. Since $\tau > t_1$ is arbitrary we get
$d_{M^K_B}(t) = \infty$ for a.e. $t
> t_1$. By | -0.285489 | -1.359376 | -0.398132 | 1.950112 | [
-0.5063494072690558,
-0.0646290814169555
] | [
-1.3737009357134409,
-1.3450508784124853
] | [
-0.47960868044792326,
-0.316655194566984
] | [
1.8772739532296423,
2.022950688251804
] | 1,024 | [
512,
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2648
] | 1,332,863 | 52,224 | ArXiv | -1 | -1 |
Theorem \ref{IV.10}(ii)
the operator $\wt A^{ac}E_{\wt A}([t_1,\infty))$ is
unitarily equivalent to $A_0E_{A_0}([t_1,\infty))$.
(ii) If $\tau \in (t_0, t_1)$, then ${\rm dim}(E_T([0,\tau))\cH) =:
p(\tau)<\infty.$ Hence, ${\rm dim}(Q\cH) \le p(\tau)$ which shows that
$d_{M_B^K}(t) \ge p(\tau)$ for a.e. $t \in (\tau, t_1)$. Since
$\tau$ is arbitrary, we obtain $d_{M_B^K}(t) \ge p(\tau)$ for
a.e. $t \in [0,t_1)$. Using Theorem \ref{IV.10}(i) we prove (ii).
(iii) By Lemma \ref{VII.1} the invariant maximal normal
function $\gotm^+(t)$ is finite for $t \in \R$. By Theorem
\ref{V.5} $\wt A^{ac}$ and $(A^F)^{ac}$ are unitarily equivalent.
Similarly we prove that $\wt A^{ac}$ and $(A^K)^{ac}$ are
unitarily equivalent. To complete the proof it remains to
apply Theorem \ref{prop7.2A}(iv).
\end{proof}
\begin{corollary}\Label{VI.5Z}
{ Let the assumptions of Theorem \ref{VII.3} be satisfied.
If ${\rm dim}(\cH) = \infty$ and $t_0:=\inf{\sigma}(T) =
\inf{\sigma}_{{\rm ess\,}}(T)=:t_1$, then
the Friedrichs extension $A^F$ is strictly $ac$-minimal.}
\end{corollary}
\begin{remark} {\em
Let ${\rm dim}(E_T([t_0, t_1))\cH) = \infty$.
{ Then there are self-adjoint extensions $\wt A= \wt A^*\in
{\rm Ext\,}_A$ of $A$ such that ${\sigma}_{ac}(\wt A) = {\sigma}(A^F) = {\sigma}_{ac}(A^F)$ but
$\wt A$ is not unitarily equivalent to $A^F$. }
}\end{remark}
\subsection{Unbounded operator potentials}
In this subsection we consider the differential expression
\eqref{8.1} with unbounded $T = T^* \ge 0,$\ $T\in \cC(\cH),$
\begin{equation}\label{8.10}
(\cA_T f)(x) = -\frac{d^2}{dx^2} f(x) + Tf(x).
\end{equation}
The minimal operator $A :=A_{T, \min}$ is defined as the closure
of the operator $A'_T$ generated on $\mathfrak H:= L^2(\R_+, \cH)$
by expression \eqref{8.10} on the domain
\begin{equation} \label{8.11}
\cD'_0 := \left\{\sum_{1\le j \le k}\phi_j(x)h_j:\
\phi_j \in W_0^{2,2}(\R_+),\;
h_j \in {\rm dom\,}(T), \;\ k \in \N \right\},
\end{equation}
that is $A'_Tf=\cA_T f,\ {\rm dom\,}(A'_T)=\cD'_0.$ Clearly, $A$ is
non-negative, since $T\ge0$ and $A_{T,min} :=
\overline{A'_T}$ coincides with $A$ defined by \eqref{8.1}
provided that $T$ is bounded.
Let $\cH_T$ be the Hilbert space which is obtained equipping the set
${\rm dom\,}(T)$ with the graph norm of $T$.
Following \cite{LioMag72} we introduce the Hilbert spaces $W^{k,2}_T({\R}_+;\cH) :=
W^{k,2}({\R}_+;\cH) \cap L^2({\ | -0.239948 | -1.139129 | -0.061763 | 2.069775 | [
0.040151088901449666,
-0.5200461615590434
] | [
-0.9260437903860113,
-1.3522133927377242
] | [
0.25273091563178446,
-0.3762568586635143
] | [
2.199843866493,
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] | 1,024 | [
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] | 1,332,863 | 53,248 | ArXiv | -1 | -1 |
R}_+,\cH_T)$, $k \in \N$, equipped with the
Hilbert norms
\begin{displaymath} \|f\|^2_{W^{k,2}_T}
=\int_{{\R}_+}\bigl(\|f^{(k)}(t)\|^2_{\cH} + \|f(t)\|^2_\cH +\|T
f(t)\|_{\cH}^2\bigr)dt.
\end{displaymath}
Obviously, we have $W^{2,2}_{0,T}(\R_+,\cH) := \{f\in
W^{2,2}_T({\R}_+;\cH):\ f(0)=f'(0) =0\} \subseteq
{\rm dom\,}(A_{T,\min})$.
\begin{lemma}\Label{VI.2.7}
{
{ Let $T=T^*$ be a non-negative operator in $\cH$. Then}
${\rm dom\,}(A_{T,\rm min})$ and $W^{2,2}_{0,T}(\R_+,\cH)$ {
coincide algebraically and topologically.}
}
\end{lemma}
\begin{proof}
{
Obviously, { for any $f \in \cD'_0$ we have}
\begin{eqnarray*}
\lefteqn{
\left\|\cA_Tf\right\|^2_\gotH
= \int_{\R_+}\left\|f''(x)\right\|^2_\cH dx } \\
& &
+ \int_{\R_+}\|Tf(x)\|_\cH ^2dx - 2{\rm Re\,}\left\{\int_{\R_+}
\left(f''(x),Tf(x)\right)_\cH dx\right\}.
\end{eqnarray*}
Integrating by parts we find
\begin{displaymath}
\int_{\R_+} \left(f''(x),Tf(x)\right)dx =
-\int_{\R_+}\left\|\sqrt{T}f'(x)\right\|^2_\cH dx.
\end{displaymath}
Combining these equalities yields
\begin{equation*}
\left\|\cA_Tf\right\|^2_\gotH =
\int_{\R_+}\left\|f''(x)\right\|dx +
\int_{\R_+}\|Tf(x)\|^2dx +
2\int_{\R_+}\left\|\sqrt{T}f'(x)\right\|^2_\cH dx
\end{equation*}
for any $f \in \cD'_0$. { Hence }
\begin{displaymath}
\|f\|^2_{W^{2,2}_T} \le \|\cA_Tf\|^2_\gotH + \|f\|^2, \quad f \in
\cD'_0.
\end{displaymath}
Furthermore, { by the Schwartz inequality,}
\begin{displaymath}
2\left|{\rm Re\,}\left\{\int_{\R_+} \left(f'(x),Tf(x)\right)_\cH dx\right\}\right| \le
\|f\|^2_{W^{2,2}_T}, \quad f \in \cD'_0.
\end{displaymath}
which gives
\begin{displaymath}
\|\cA_Tf\|^2_\gotH + \|f\|^2 \le 2\|f\|^2_{W^{2,2}_T}, \quad f \in \cD'_0.
\end{displaymath}
{ Thus, we arrive at the two-sided estimate }
\begin{displaymath}
\|f\|^2_{W^{2,2}_T} \le \left\|\cA_Tf\right\|^2_\gotH +
\|f\|^2_\gotH \le 2\|f\|^2_{W^{2,2}_T}\qquad f \in \cD'_0.
\end{displaymath}
{ Since $\cD'_0$ is dense in
$W^{2,2}_{0,T}$ too, we obtain that ${\rm dom\,}(A_{T,\rm
min})$ coincides with $W^{2,2}_{0,T}$ algebraically and topologically.}
}
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] | 1,332,863 | 54,272 | ArXiv | -1 | -1 |
end{proof}
{ Since $A$ is non-negative it admits the Friedrichs
extension $A^F$ and the Krein extension $A^K$.
We define the
extension $A^N$ as the self-adjoint operator associated with the
closed quadratic form ${\mathfrak t}_N$,
\begin{equation}
{\mathfrak t}_N[f] := \int^\infty_0 \left\{\|f'(x)\|^2_\cH +
\|{\sqrt T}f(x)\|^2_\cH\right\}dx = \|u\|^2_{W^{1,2}_{\sqrt T}} - \|u\|^2_{L^2(\R_+,\cH)} ,
\end{equation}
$ {\rm dom\,}({\mathfrak t}_N) := W^{1,2}_{\sqrt T}(\R_+,\cH)$.
The definition makes sense for $T\in [\cH]$. In this case
$A^N=A_1$ with $A_1$ defined in Theorem \ref{prop7.2A}(iii). }
{ We also put $\gt_F := \gt_N\upharpoonright{\rm dom\,}(\gt_F)$,
${\rm dom\,}(\gt_F) := \{f\in W^{1,2}_{\sqrt{T}}(\R_+;\cH):\ f(0)=0\}. $}
\begin{proposition}\label{prop6.8}
Let $T=T^*\in\cC(\cH)$, $T\ge 0$, and { let} $A:=A_{T,\min}$
{ be } defined by \eqref{8.10}-\eqref{8.11}. Let also
$\cH_n:={\rm ran\,}\bigl(E_T([n-1,n))\bigr)$, $T_n:= T E_T([n-1,n))$ and
let $S_n$ { be} the closed minimal symmetric operator defined
by \eqref{8.1} in $\gH_n:=L^2({\mathbb R}_+,\cH_n)$ with $T$
replaced by $T_n.$ Then
\item[\rm\;\;(i)] { the following decompositions hold}
\begin{equation}\label{8.23}
A = \bigoplus^\infty_{n=1} S_n, \quad A^F =
\bigoplus^\infty_{n=1} S^F_n, \quad A^K = \bigoplus^\infty_{n=1}
S^K_n, \quad A^N = \bigoplus^\infty_{n=1}S^N_n.
\end{equation}
\item[\rm\;\;(ii)] { The domain ${\rm dom\,}(A^F)$ equipped with the
graph norm is a closed subspace of $W^{2,2}_T(\R_+,\cH),$}
\begin{displaymath}
{\rm dom\,}(A^F) = \{f \in W^{2,2}_T(\R_+,\cH): f(0) = 0\}.
\end{displaymath}
\item[\rm\;\;(iii)] { The domain ${\rm dom\,}(A^N)$ equipped with the
graph norm is a closed subspace of $W^{2,2}_T(\R_+,\cH),$}\
${\rm dom\,}(A^N) = \{f \in W^{2,2}_T(\R_+,\cH): f'(0)=0\}$.
\end{proposition}
\begin{proof}
{ (i)} We introduce the sets
\begin{displaymath}
\cD''_0 := \left\{\sum_{1\le j \le k}\phi_j(x)h_j:\
\phi_j \in W_0^{2,2}(\R_+),\;
h_j \in \cH_n, \; k,n \in \N \right\}
\end{displaymath}
and $\cD''_{0n} := \left\{f \in \cD''_0: f(x) \in \cH_n\right\},
\quad n \in \N.$ Obviously, we have
$\cD''_0 = \bigoplus^\infty_{n=1}\cD''_{0n} \subseteq \cD'_0$.
| 0.074693 | -1.010203 | 0.389688 | 2.241466 | [
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] | 1,332,863 | 55,296 | ArXiv | -1 | -1 |
Setting $A''_T := A'_T\upharpoonright\cD''_0$ we find
$\overline{A''_T} = \overline{A'_T} = A_{T,\rm min}$. Moreover,
setting $A''_n := A_n\upharpoonright\cD''_{0n}$, $n \in \N$, we have
$\overline{A''_n} = A_n$, $n \in \N$. Since
$A''_T = \bigoplus^\infty_{n=1}A''_n \subseteq A'_T,$
we obtain
\begin{displaymath}
A_{T,min} = \overline{A''_T} = \bigoplus^\infty_{n=1}\overline{A''_n}
= \bigoplus^\infty_{n=1}A_n \subseteq A_{T,min}
\end{displaymath}
which proves { the first relation of \eqref{8.23}. The second
and the third relations are implied by Corollary \ref{cor5.5}}.
{ To prove the last relation of \eqref{8.23} we set $S^N :=
\bigoplus^\infty_{n=1}S^N_n.$ Since $S^N_n= (S^N_n)^*\in
{\rm Ext\,}_{S_n}$ and $A = \oplus^\infty_{n=1} S_n,$ $S^N$ is a
self-adjoint extension of $A,\ S^N\in {\rm Ext\,}_A.$} Let
$f=\oplus^\infty_{n=1} f_n\in {\rm dom\,}(S^N).$ { Then integrating
by parts we obtain}
\begin{eqnarray*} \lefteqn{ (S^Nf,f) = \sum_1^{\infty}(S^N_nf_n, f_n) =
\sum_{n=1}^{\infty} \int^\infty_0 \left\{\|f'_n(x)\|^2_{\cH_n} +
\|{\sqrt T_n}f_n(x)\|^2_{\cH_n}\right\}dx} \\
& & = \int^\infty_0 \left\{\|f'(x)\|^2_{\cH} +
\|{\sqrt T}f(x)\|^2_{\cH}\right\}dx
= {\mathfrak t}_N[f].
\end{eqnarray*}
{ Since, by definition, $A^N$ is associated with the
quadratic form ${\mathfrak t}_N,$ the last equality yields
$S^N\subset A^N.$ Hence $S^N = A^N,$ since both $S^N$ and $A^N$
are self-adjoint extensions of $A$.}
{
(ii)\ Following the reasoning of Lemma \ref{VI.2.7} we find
\begin{equation}\Label{8.23b}
\|f_n\|^2_{W^{2,2}_{T_n}} \le \|S^F_n f_n\|^2_{\gotH_n} +
\|f_n\|^2_{\gotH_n} \le 2\|f_n\|^2_{W^{2,2}_{T_n}}, \qquad n \in
\N,
\end{equation}
{ where} $f_n \in {\rm dom\,}(S^F_n) = \{g_n \in
W^{2,2}(\R_+,\cH_n): g_n(0) =0\}$. { Using
representation \eqref{8.23} for $A^F$ and setting $f^m :=
\oplus^m_{n=1} f_n$, $f_n \in {\rm dom\,}(F_n)$, we obtain from
\eqref{8.23b}}
\begin{equation}\Label{8.23a}
\|f^m\|^2_{W^{2,2}_T} \le \|A^F f^m\|^2_{\gotH} +
\|f^m\|^2_{\gotH} \le 2\|f^m\|^2_{W^{2,2}_T}, \quad m \in \N.
\end{equation}
{ Since the set $\{f^m = \oplus^m_{n=1} f_n: \ f_n \ | -0.269396 | -1.384445 | -0.074444 | 2.033356 | [
-0.4556714163961016,
-0.08311969970843877
] | [
-1.4954636792425018,
-1.2734257351600966
] | [
-0.20632870996276828,
0.057440356677195714
] | [
1.9293013585947,
2.1374109800549306
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] | 1,332,863 | 56,320 | ArXiv | -1 | -1 |
in
{\rm dom\,}(S^F_n),\ m \in \N \}$, is a core for $A^F,$ inequality
\eqref{8.23a} remains valid for} $f \in {\rm dom\,}(A^F)$. This shows
that ${\rm dom\,}(A^F) = \{f \in W^{2,2}_T(\R_+,\cH): f(0) = 0\}$. {
Moreover, due to \eqref{8.23a} the graph norm of $A^F$ and the
norm $\|\cdot\|_{W^{2,2}_T}$ restricted to ${\rm dom\,}(A^F)$ are
equivalent}.
(iii)\ { Similarly to \eqref{8.23b} one gets}
\begin{displaymath}
\|f_n\|^2_{W^{2,2}_{T_n}} \le \|S^N_n f_n\|^2_{\gotH_n} + \|f_n\|^2 \le 2\|f_n\|^2_{W^{2,2}_{T_n}}
\end{displaymath}
for $f_n \in {\rm dom\,}(S^N_n) = \{g_n \in W^{2,2}(\R_+,\cH_n): g'_n(0)
= 0\}$, $n \in \N$. { It remains to repeat the reasonings of
(ii)} }
\end{proof}
To { extend} Theorem \ref{prop7.2A} { to the case of
unbounded operators $T=T^*\ge0$} we first construct a boundary
triplet for $A^*$, using { Theorem \ref{VI.3} and
representation \eqref{8.23} for} $A.$
\begin{lemma}\label{lem6.9}
{
Assume conditions of Proposition} \ref{prop6.8}. Then there is a
sequence of boundary triplets $\wh{\Pi}_n =
\{\cH_n,\wh{\Gamma}_{0n},\wh{\Gamma}_{1n}\}$ { for $S_n^*$} such that
${\Pi} := \oplus^\infty_{n=1}\wh{\Pi}_n =: \{\cH,\wh
{\Gamma}_0,\wh {\Gamma}_1\}$ { forms an ordinary boundary triplet for} $A^*.$
{ Moreover, $A^F = A^*\upharpoonright{\xker\,}(\wh {\Gamma}_0)$ and
the corresponding Weyl function is}
\begin{equation}\Label{8.7}
\wh M(z) = \frac{i\sqrt{z-T}
+\im(\sqrt{i - T})}{\re(\sqrt{i-T})}.
\qquad z \in \C_+,
\end{equation}
\end{lemma}
\begin{proof}
For any $n \in \N$ we define a boundary triplet ${\Pi}_n =
\{\cH_n,{\Gamma}_{0n},{\Gamma}_{1n}\}$ for { $S_n^*$} with
${\Gamma}_{0n},{\Gamma}_{1n}$ defined by \eqref{8.2}. By Theorem
\ref{prop7.2A}(i) { $S^F_n = S_{0n} =
S_n^*\upharpoonright{\xker\,}({\Gamma}_{0n})$} and by { Lemma
\ref{lem6.1}}
the corresponding Weyl function is $M_n(z) = i\sqrt{z-T_n}$.
Following Lemma \ref{VI.1}, cf. \eqref{3.8}, we define a sequence
of regularized boundary triplets $\wh{\Pi}_n =
\{\cH_n,\wh{\Gamma}_{0n},\wh{\Gamma}_{1n}\}$ { for $S_n^*$} by setting
$R_n:=(\re(\sqrt{i-T_n}))^{1/2},$ \ $Q_n:= -\im(\sqrt{i - T_n})$
and
\begin{equation}\label{6.22A}
\wh{\Gamma}_{0n} := R_n{\Gamma}_{0n}, \quad \wh{\Gamma}_{1n} := R_n^{-1}({\Gamma}_{1n}
- Q_n{\Gamma}_{0n | -0.084746 | -1.219707 | -0.059227 | 2.158222 | [
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] | 1,332,863 | 57,344 | ArXiv | -1 | -1 |
}), \qquad n\in \N.
\end{equation}
Hence { $S^F_n = S_{0n}$} and the corresponding Weyl
function $\wh M_n(\cdot)$ is given by
\begin{equation} \Label{8.7B}
\wh M_n(z) = \frac{i\sqrt{z-T_n}
+\im(\sqrt{i - T_n})}{\re(\sqrt{i-T_n})},
\qquad z \in \C_+, \qquad n \in \N.
\end{equation}
By Theorem \ref{VI.3} the direct sum $\wh{\Pi} :=
\bigoplus_{n=1}^\infty\wh{\Pi}_n = \{\cH,\wh{\Gamma}_{0},\wh{\Gamma}_{1}\}$
forms a boundary triplet for $A^*$ and
the corresponding Weyl function is
\begin{equation}\Label{8.7C}
\wh M(z) = \bigoplus_{n\in\N}\wh M_n(z) \qquad z \in \C_+.
\end{equation}
Combining \eqref{8.7C} with \eqref{8.7B} we arrive at
\eqref{8.7}.
{
Combining Theorem \ref{VI.3} (cf. \eqref{6.8}) with
Corollary \ref{cor5.5} we get
\begin{equation}\Label{6.25z}
A_0 = A^*\upharpoonright{\xker\,}(\wh {\Gamma}_0) =
\bigoplus^\infty_{n=1}S^*_n\upharpoonright{\xker\,}(\wh {\Gamma}_{0n}) =
\bigoplus^\infty_{n=1}S_{0n} = \bigoplus^\infty_{n=1}S^F_{n}=A^F
\end{equation}
which proves the second assertion.}
\end{proof}
{ Next we generalize Theorem \ref{prop7.2A} to the case of
unbounded operator potentials.}
\begin{theorem}\Label{VI.9}
Let $T = T^* \ge 0,$\ $t_0 := \inf{\sigma}(T)$, and $A := A_{T,min}$,
cf. \eqref{8.10}-\eqref{8.11}. { Let also $\wh {\Pi} =
\{\cH,\wh {\Gamma}_0,\wh {\Gamma}_1\}$ be the boundary triplet for $A^*$
defined by Lemma \ref{lem6.9} and $\wh M(\cdot)$ the
corresponding Weyl function (cf. \eqref{8.7}).}
{ Then}
\item[\rm \;\;(i)] The Friedrichs extension $A^F$ coincides with
$A_0 := A^*\upharpoonright{\xker\,}(\wh {\Gamma}_0).$
Moreover, $A^F$ is absolutely continuous, $A^F =
(A^F)^{ac}$, and ${\sigma}(A^F) = {\sigma}_{ac}(A^F) = [t_0,\infty)$.
\item[\rm \;\;(ii)] The Krein extension $A^K$ is given by
$A_{B^K}:=A^*\upharpoonright {\xker\,}({\Gamma}_1 - B^K{\Gamma}_0)$, where
\begin{equation}\Label{6.32x}
B^K = \frac{1}{\sqrt{2}\sqrt{T} + \sqrt{T + \sqrt{1
+ T^2}}} \frac{1}{\sqrt{T + \sqrt{1 + T^2}}}.
\end{equation}
Moreover, ${\xker\,}(A^K) = \gotH_0 := \overline{\gotH'_0}$, $\gotH'_0
:= \{e^{-x\sqrt{T}}h: h \in {\rm ran\,}(T^{1/4})\},$ the restriction $A^K
\upharpoonright{\rm dom\,}(A^K) \cap \gotH^\perp_0$ is absolutely
continuous, and $A^K = 0_{\gotH_0} \oplus (A^K)^{ac}$.
\item[\rm \;\;(iii)]
{ | 0.103156 | -1.162407 | 0.07139 | 2.158222 | [
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] | 1,332,863 | 58,368 | ArXiv | -1 | -1 |
The extension $A^N$ is given by $A^N =
A^*\upharpoonright {\xker\,}(\wh {\Gamma}_1 - B^N\wh {\Gamma}_0)$ where $B^N :=
\sqrt{T + \sqrt{1 + T^2}}$.} Moreover, $A^N$ is absolutely
continuous, $A^N = (A^N)^{ac}$ and ${\sigma}(A^N) = {\sigma}_{ac}(A^N) =
[t_0,\infty)$.
\item[\rm \;\;(iv)]
The operators $A^F$, $(A^K)^{ac}$ and and $A^N$ are unitarily
equivalent.
\end{theorem}
\begin{proof}
(i) { This statement is implied by combining Theorem
\ref{prop7.2A} with \eqref{6.25z}}.
(ii) Using the polar decomposition $i-{\lambda} = \sqrt{1 +
{\lambda}^2}e^{i\theta({\lambda})}$ with $\theta({\lambda}) = \pi - \arctan(1/{\lambda})$,
${\lambda} \ge 0$ we get
\begin{equation} \label{6.23A} \re(\sqrt{i - T}) = \int^\infty_0 \sqrt[4]{1 +
{\lambda}^2}\cos(\theta({\lambda})/2)dE_T({\lambda}). \end{equation}
Setting $\varphi({\lambda}) = \arctan(1/{\lambda})$, ${\lambda} \ge 0$ and noting
that $\cos(\varphi({\lambda})) = {{\lambda}}{(1 + {\lambda}^2})^{-1/2},$ we find
$\cos(\theta({\lambda})/2) = {2}^{-1/2}(1 + {\lambda}^2)^{-1/4}({\lambda} + \sqrt{1
+ {\lambda}^2})^{-1/2}.$ Substituting this expression in \eqref{6.23A}
{ yields}
\begin{equation}\Label{6.30a}
\re(\sqrt{i - T}) = {2}^{-1/2}(T + \sqrt{1 + T^2}\
)^{-1/2}.
\end{equation}
Similarly, taking into account $\sin(\theta({\lambda})/2) =
\cos(\varphi({\lambda})/2)$ and $\cos(\varphi({\lambda})/2) = {2}^{-1/2}(1 +
{\lambda}^2)^{-1/4}({\lambda} + \sqrt{1 + {\lambda}^2})^{1/2}$, we get
\begin{equation}\Label{6.35}
\im(\sqrt{i- T}) = \int^\infty_0 \sqrt[4]{1 +
{\lambda}^2}\cos(\varphi({\lambda})/2)dE_T({\lambda}) = \frac{1}{\sqrt{2}}\sqrt{T + \sqrt{1 + T^2}}.
\end{equation}
{ It follows from \eqref{8.7} with account of \eqref{6.30a} and \eqref{6.35} that}
$M(0):=\,\mbox{\rm s-}\hspace{-2pt} \lim_{x\to+0}M(-x) =: B^K$ { where $B^K$ is} defined
by \eqref{6.32x}. Therefore, by \cite[Proposition 5(iv)]{DM91} the
Krein extension $A^K$ is given by $A_{B^K}:=A^*\upharpoonright
{\xker\,}({\Gamma}_1 - B^K{\Gamma}_0).$ { The second statement follows from Proposition \ref{prop6.8} and
Theorem \ref{prop7.2A}(ii).}
(iii) It is easily seen that in the boundary triplet $\wh {\Pi}_n
= \{\cH_n,\wh {\Gamma}_{0n},\wh {\Gamma}_{1n}\}$ { defined by
\eqref{6.22A}} the extension $A^N_n$ admits a representation
$A^N_n = A_{B_n}$ where $B_n := \sqrt{T_n + \sqrt{1 + T^2_n}}$,
$n \in \N$. By Proposition \ref{prop6. | -0.23327 | -1.294913 | -0.221546 | 1.976126 | [
-0.27008039576676957,
-0.1964603414580862
] | [
-1.2805882494853356,
-1.309238306786291
] | [
-0.1226327561250874,
-0.3204595561050604
] | [
2.0749780936168616,
1.8772739532296423
] | 1,024 | [
512,
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] | 1,332,863 | 59,392 | ArXiv | -1 | -1 |
8}, $A^N =
\oplus^\infty_{n=1}A^N_n = A_{B^N}$ where $B^N =
\oplus^\infty_{n=1}B_n$. The remaining part of (iii) follows from
the representation $A^N = \bigoplus^\infty_{n=1}A^N_n$ and Theorem
\ref{prop7.2A}(iii).
(iv) { The assertion follows from Theorem \ref{prop7.2A}(iv)
and \eqref{8.23}.}
\end{proof}
{ Next we generalize Theorem \ref{VII.3} to the case of
unbounded $T\ge0.$}
\begin{theorem}\Label{VI.10a}
Let $T = T^* \ge 0$ and $t_1 := \inf{\sigma}_{\rm ess}(T)$.
Further, let $A := A_{T,\rm min}$, cf. \eqref{8.10}-\eqref{8.11},
and $\wt A = \wt A^* \in Ext_A$. Then
\item[\rm\;\;(i)] the absolutely continuous part
$\wt A^{ac} E_{\wt A}([t_1,\infty))$ is unitarily
equivalent to the part $A^FE_{A^F}([t_1,\infty)) =
(A^F)^{ac}E_{A^F}([t_1,\infty))$;
\item[\rm\;\;(ii)] the Friedrichs extension $A^F$ is
$ac$-minimal { and} $\sigma_{ac}(A^F) \subseteq
\sigma_{ac}(\wt A);$
\item[\rm \;\;(iii)] the $ac$-part $\wt A^{ac}$ of $\wt A$
is unitarily equivalent to $A^F$ if either $(\wt A - i)^{-1} -
(A^F - i)^{-1} \in \gotS_\infty(\gotH)$ or $(\wt A - i)^{-1} -
(A^K - i)^{-1} \in \gotS_\infty(\gotH)$.
\end{theorem}
\begin{proof}
By Corollary \ref{V.1A} it suffices to assume that the
extension $\wt A = \wt A^*$ is disjoint with $A_0,$ that is,
it admits a representation $\wt A = A_B$ with $B\in \cC(\cH).$
(i) { We} consider the boundary triplet $\wh {\Pi} = \{\cH,\wh
{\Gamma}_0,\wh {\Gamma}_1\}$ defined { in Lemma \ref{lem6.9}}. By
Proposition \ref{prop3.1} the generalized Weyl function
corresponding to the generalized boundary triplet $\wh {\Pi}_B$
{ is defined by} $\wh M_B(z) = (B - \wh M(z))^{-1}$, $z \in
\C_+$, where $\wh M(z)$ is given by \eqref{8.7}. Clearly,
\begin{equation}\Label{6.32AB}
\im(\wh M_B(z)) = \wh M_B(z)^*\im(\wh M(z))\wh M_B(z), \qquad z
\in \C_+.
\end{equation}
{ It follows from \eqref{8.7} that $\bigl(\re(\sqrt{i -
T})\bigr)^{-1}\ge \sqrt 2.$ Therefore \eqref{6.30a} yields}
\begin{equation}\Label{6.32a}
\im(\wh M(z)) \ge \sqrt{2}\im(M(z)), \quad z \in
\C_+, \quad \text{where} \quad M(z) = i\sqrt{z - T},
\end{equation}
cf. \eqref{8.2B}. Following the line of reasoning of the proof of
Theorem \ref{VII.3}(i) we obtain from \eqref{6.32a} that $d_{\wh
M^D}(t)=\infty$ for a.e. $t\in [t_1, \infty),$ where $D=D^*\in
\gotS_2(\cH)$ and ${\xker\,} D=\{0\}.$ Moreover | -0.283777 | -1.388026 | -0.09093 | 1.976126 | [
-0.5885299330089814,
0.020975632895467018
] | [
-1.25193819218438,
-1.5241137365434572
] | [
-0.11756027407431886,
-0.0642992125412492
] | [
1.8356520289375962,
2.1166000179089077
] | 1,024 | [
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12808
] | 1,332,863 | 60,416 | ArXiv | -1 | -1 |
, it follows from
\eqref{6.32AB} that $d_{\wh M_B^D}(t)= d_{\wh M^D}(t)=\infty$ for
a.e. $t\in [t_1, \infty).$ One completes the proof by applying
Theorem \ref{IV.10}.
(ii) To prove (ii) we use again estimates \eqref{6.32a} and
follow the proof of Theorem \ref{VII.3}(ii). We complete the proof
by applying Theorem \ref{IV.10}.
(iii) The Weyl function $\wh M(\cdot)$
is given by \eqref{8.7}. { Taking into account
\eqref{8.7C} one obtains $\sup_{n\in\N}\gotm^+_n < \infty$,
where $\gotm^+_n$ is the maximal normal invariant function defined by
\eqref{4.12A}. Indeed, this follows from
\eqref{7.3a} because this estimate shows that $\gotm^+_n$ does not
depend on $n\in \N$. Applying Theorem \ref{V.5} and Remark \ref{V.7}
we complete the proof.}
{ To prove the second statement we note that the operator
$B^K$ defined by \eqref{6.32x} is bounded}. { Therefore, by
Proposition \ref{prop3.1} } a triplet $\wh {\Pi}_{B^K} := \{\cH,\wh
{\Gamma}^{B^K}_0,\wh {\Gamma}^{B^K}_1\}$ with $\wh {\Gamma}^{B^K}_1 := \wh
{\Pi}_0$ and $\wh {\Gamma}^{B^K}_0 := B^K\wh {\Gamma}_0 - \wh {\Gamma}_1$, is a
boundary triplet for $A^*$ such that $A^K :=
A^*\upharpoonright{\xker\,}(\wh {\Gamma}^{B^K}_0)$. The corresponding Weyl
function is
\begin{displaymath}
{ \wh M_{B^K}(z) = (B^K - \wh M(z))^{-1}}, \quad z \in \C_+.
\end{displaymath}
Inserting expression \eqref{6.32x} into this formula we get
\begin{displaymath} { \wh M_{B^K}(z)} = -\frac{1}{\sqrt{2}}\frac{1}{\sqrt{T} +
i\sqrt{z-T}}\frac{1}{\sqrt{T + \sqrt{1 + T^2}}}=
\frac{1}{z\sqrt{2}}\frac{\sqrt{T} - i\sqrt{z - T}}{\sqrt{T +
\sqrt{1 + T^2}}}.
\end{displaymath}
It follows that the limit ${ \wh M_{B^K}(t+i0)}$ exists for any
$t\in \R \setminus \{0\}$ and
\begin{displaymath}
{ \wh M_{B^K}(t)} := \,\mbox{\rm s-}\hspace{-2pt} \lim_{y\to*0}M_{B^K}(t+iy) =
-\frac{1}{t\sqrt{2}}\frac{\sqrt{T} - i\sqrt{t - T}}{\sqrt{T + \sqrt{1 + T^2}}}.
\end{displaymath}
Clearly, ${ \wh M_{B^K}(t)}\in [\cH]$ for any $t\in \R \setminus \{0\}.$
{ By Theorem \ref{V.5} the $ac$-parts of $\wt A$} and $A^K$
are unitarily equivalent whenever $(\wt A - i)^{-1} - (A^K -
i)^{-1} \in \gotS_\infty(\gotH).$ This completes the {
proof.}
\end{proof}
{ Finally, we generalize Corollary \ref{VI.5Z} to unbounded
operator potentials.}
\begin{corollary}\label{cor6.12}
{ Assume conditions of Theorem \ref{VI.10a}. If ${\rm dim}(\cH) =
\infty$ and $t_0:=\inf{\sigma}(T) = \inf{\sigma}_{{\rm ess\,}}(T)=:t_1$, then the
Fried | -0.099128 | -1.42742 | -0.183503 | 1.976126 | [
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] | 1,332,863 | 61,440 | ArXiv | -1 | -1 |
richs extension $A^F$ and the Krein extension $A^K$ are
strictly $ac$-minimal.}
\end{corollary}
\subsection{Application}
In this subsection we apply previous results of this section to
Schroedinger operator in the half-plane. To this end we denote by
$L = L_{\rm min}$ the minimal elliptic operator associated in
$L^2(\Omega), \ \Omega:=\R_+\times \R^n,$ with the differential
expression
\begin{equation}\label{7.40}
\mathcal L := -\Delta + q(x) := -\bigl(\frac{\partial^2}{\partial
t^2} +\sum_{j=1}^n\frac{\partial^2}{\partial x_j^2}\bigr) + q(x),
\quad (t,x)\in \Omega,
\end{equation}
where $q= \bar q \in L^\infty(\R),$ \ $x := (x_1, \ldots, x_n)$
and $n\ge 1.$
Recall that $L_{\min}$ is the closure of $\mathcal L$,
defined on $C^\infty_0(\Omega).$
It is known that ${\rm dom\,}(L_{\min}) = H^{2}_0(\Omega).$ Clearly, $L$
is symmetric. The maximal operator $L_{\max}$ is then defined by
$L_{\max} = (L_{\min})^*.$ We emphasize that $H^{2}(\Omega)\subset
{\rm dom\,}(A_{\max})\subset H^{2}_{loc}(\Omega)$, while
${\rm dom\,}(L_{\max})\not = H^{2}(\Omega)$.
Next we define the trace mappings $\gamma_j\colon
C^{\infty}({\overline{\Omega}})\to C^{\infty}(\partial\Omega), \
j\in \{0,1\},$ by setting $\gamma_0 u := u \upharpoonright
{\partial\Omega}$ and $\gamma_1 u := \gamma_0
({\partial u}/{\partial n})$
where $n$ stands for the interior normal to $\partial\Omega$.
Denote by $D_{\mathcal L}(\Omega)$ the domain ${\rm dom\,}(L_{\max})$
equipped with the graph norm. It is known (see \cite{LioMag72,
Gru08}) that $\gamma_j$ can be extended by continuity to the
operators mapping $D_{\mathcal L}(\Omega)$ continuously onto
$H^{-j-1/2}(\partial\Omega), \ j\in \{0,1\}.$
Let us define the
following extensions of $L_{\min}$ (realizations of $\mathcal
L$):
\begin{enumerate}
\item[(i)] $L^Df := \mathcal L[f],$ \ $f \in {\rm dom\,} (L^D) :=
\{\varphi \in
H^2(\R_+\times \R):\ \gamma_0 \varphi = 0\}$;
\item[(ii)] $L^Nf := \mathcal L [f],$\ $f \in {\rm dom\,} (L^N) :=
\{\varphi \in
H^2(\R_+\times \R): \;
\gamma_1 \varphi=0\}$;
\item[(iii)] $L^Kf := \mathcal L[f],$\ $f \in {\rm dom\,} (L^K) :=
\{\varphi \in {\rm dom\,}(A_{\max}): \gamma_1 \varphi + \Lambda \gamma_0
\varphi = 0\}, \quad \text{where} \quad \Lambda := \sqrt{
-\Delta_x + q(\cdot)}:\ H^{-1/2}(\partial\Omega)\to
H^{-3/2}(\partial\Omega).$
\end{enumerate}
To treat the operator $L_{\min}$ as the Sturm-Liouville operator
with (unbounded) operator potential we denote by $T$ the (closed)
minimal operator associated in $\cH := L^2({\mathbb R}^n)$ with
the Schr\"odinger expression
\begin{equation}\label{7.41}
-\Delta_x + q(x) := - \sum_{j=1}^n\frac{\partial^2}{\partial
x_j^2} | -0.033469 | -1.019156 | 0.103093 | 2.246669 | [
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] | 1,332,863 | 62,464 | ArXiv | -1 | -1 |
+ q(x).
\end{equation}
Since $q= \bar q \in L^\infty(\R),$ the operator $T$ is
self-adjoint and semibounded. Moreover, $T\ge 0$ if $q(\cdot)\ge0.$
Let $A := A_{T,\rm min}$ be the minimal operator
associated with \eqref{8.10} where $T$ is defined by \eqref{7.41}.
\begin{proposition}\label{prop6.12}
Let $q(\cdot) \in L^\infty(\R),$ \ $q(\cdot) \ge 0$, and let $T$
be the minimal (self-adjoint) operator associated in $L^2(\R)$ with
the Schr\"odinger expression \eqref{7.41}. Let also $t_0:=\inf
\sigma(T)$ and $t_1:=\inf \sigma_{{\rm ess\,}}(T).$
Then:
\item[\rm \;\;(i)] the operator $A_{T,\rm min}$ coincides with
$L=L_{\rm min}$ and ${\rm dom\,}(A_{T,\rm min})= H^2_0(\Omega);$
\item[\rm\;\;(ii)] the Friedrichs extension $A^F$ coincides with
$L^D$, hence $L^D$ is absolutely continuous,
$\sigma(L^D)=\sigma_{ac}(L^D) = [t_0,\infty)$ and
$N_{L^D}(t)=\infty$ for a.e. $t\in [t_0,\infty);$
\item[\rm\;\;(iii)] the Krein extension $A^K$ coincides with
$L^K$, in particular, $L^K$ admits the decomposition $L^K =
0_{\cH_0}\oplus (L^K)^{ac}$, $\cH_0 := {\xker\,}(L^K)$, and
$\sigma_{ac}(L^K)=[t_0,\infty)$;
\item[\rm\;\;(iv)] the extension $A^N$ defined by \eqref{8.23},
coincides with $L^N$, in particular, $L^N$ is absolutely
continuous and $ \sigma(A^N) = \sigma_{ac}(A^N)=[t_0, \infty);$
\item[\rm\;\;(v)] the operators $L^D$,\ $L^N,$\ and $(L^K)$ are
$ac$-minimal, in particular, $L^D$,\ $L^N,$\ and
$(L^K)^{ac}$ are pairwise unitarily equivalent. If, in addition,
$t_0=t_1,$ then the operators $L^D$,\ $L^N,$\ and $(L^K)$ are
strictly $ac$-minimal;
\item[\rm\;\;(vi)] if ${\wt L}$
is a self-adjoint extension of $L$ and $({\wt
L}-i)^{-1}-(L^D-i)^{-1}\in {\mathfrak S}_{\infty},$ then ${\wt
L}^{ac}$ and $L^D$ are unitarily equivalent. If ${\wt L}$
satisfies $({\wt L}-i)^{-1}-(L^K-i)^{-1}\in {\mathfrak
S}_{\infty}$, then ${\wt L}^{ac}$ and $L^D$ are unitarily
equivalent.
\end{proposition}
\begin{proof}
(i) We introduce the set
\begin{displaymath} \cD'_\infty := \left\{\sum_{1\le j \le k} \phi_j(x)h_j(\xi):
\;\phi_j \in C^\infty_0(\R_+), \ h_j \in C^\infty_0(\R),\ k \in
\N\right\} \end{displaymath}
We note that $\cD'_\infty \subseteq \cD'_0$ and $\cD'_\infty
\subseteq C^\infty_0(\R_+\times\R)$. Moreover, $A_{T,\rm min}
\upharpoonright\cD'_\infty = L\upharpoonright\cD'_\infty$. Since
$\c | -0.284462 | -1.31282 | 0.049832 | 2.064573 | [
-0.0749016471344462,
-0.49402232840806687
] | [
-1.3951884786891575,
-1.2304506492086633
] | [
0.0751940438548856,
0.024469223347200215
] | [
2.0541671314708383,
2.0749780936168616
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] | 1,332,863 | 63,488 | ArXiv | -1 | -1 |
D'_\infty$ is a core for both $A_{T,\rm min}$ and $L_{\rm
min},$ we have $A_{T,\rm min} = L_{\rm min}$.
It is clear (after applying the Fourier transform) that ${\rm dom\,}(T)
={\rm dom\,}(\Delta_x) = H^2(\R^n)$. Therefore, by Lemma \ref{VI.2.7},
${\rm dom\,}(A_{T,\rm min})= W^{2,2}_{0,T}=H^2_0(\Omega).$
\noindent (ii) Since ${\rm dom\,}(T) = H^2(\R^n),$ we have
$W^{2,2}_{T}(\R_+;\cH) = H^2(\Omega).$ Therefore by Proposition
\ref{prop6.8} $A^F=L^D.$ The second assertion follows from Theorem
\ref{VI.9}(i).
\noindent (iii)\ It is proved in \cite[Section 9.7]{DM91} that
$L^K$ is the Krein extension of $L_{\min}.$ The rest of the
statements is implied by Theorem \ref{VI.9}(ii).
\noindent (iv) The equality $A^N = L^N$ is immediate from
Proposition \ref{prop6.8}(iii). The second statement follows from
Theorem \ref{VI.9}(iii).
\noindent (v) By Theorem \ref{VI.10a}(ii) the extension
$L^D(=A^F)$ is $ac$-minimal. By Theorem \ref{VI.9}(iv) $L^D$,\
$L^N (=A^N),$\ and $(L^K)^{ac} (= (A^K)^{ac})$ are pairwise
unitarily equivalent, hence $L^N,$\ and $L^K$ are $ac$-minimal
too. The last statement is immediate from Corollary \ref{cor6.12}.
\noindent (vi) The statement is immediate from
(ii), (iii) and Theorem \ref{VI.10a}(iii).
\end{proof}
\begin{remark}
{\em
Let $T$ be the (closed) minimal non-negative operator associated
in $\cH := L^2({\mathbb R}^n)$ with general uniformly elliptic
operator
\begin{equation}
- \sum_{j,k=1}^n \frac{\partial}{\partial x_j} a_{jk}(x)
\frac{\partial}{\partial x_j} + q(x), \quad a_{jk} \in
C^{1}({\overline\Omega}),\ \ q \in C({\overline\Omega})\cap
L^\infty({\Omega}),
\end{equation}
where the coefficients $a_{jk}(\cdot)$ are bounded with their
$C^1$-derivatives, $q\ge 0.$ If the coefficients have some
additional "good" properties, then ${\rm dom\,}(T)=H^2({\R}^n)$
algebraically and topologically. By Lemma \ref{VI.2.7},
{ ${\rm dom\,}(A_{T,\min})= W^{2,2}_{0,T}(\R_+,\cH) =
H^{2,2}_0(\Omega)$ } and Proposition \ref{prop6.12} remains valid with
$T$ in place of the Schr\"odinger operator \eqref{7.41}.
Note also that the Dirichlet and the Neumann realizations $L^D$
and $L^N$ are always self-adjoint ((cf. \cite[Theorem
2.8.1]{LioMag72}, \cite{Gru08})).
}
\end{remark}
\begin{corollary}\label{cor6.13}
Assume the conditions of Proposition \ref{prop6.12}. If, in
addition,
\begin{equation}\label{6.33}
\lim_{|x|\to\infty}\int_{|x-y|\le 1}|q(y)|dy = 0,
\end{equation}
then the operators $L^D$,\ $L^N,$\ and $(L^K)$ are strictly
$ac$-minimal,
\begin{displaymath}
{\sigma}(L^D) = {\ | 0.176776 | -0.979763 | 0.338963 | 2.210249 | [
-0.008472388828006331,
0.3620248147161584
] | [
-0.7648872180681365,
-1.194638077582469
] | [
0.3846154489517664,
0.29331077203793277
] | [
2.158221942200954,
2.262276752931069
] | 1,024 | [
512,
512
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2802
] | 1,332,863 | 64,512 | ArXiv | -1 | -1 |
sigma}_{ac}(L^D) = {\sigma}_{ac}(L^K) = {\sigma}(L^N) ={\sigma}_{ac}(L^N)
= [0,\infty),
\end{displaymath}
and $N_{E_{L^D}}(t) = N_{E_{L^N}}(t)= N_{E^{ac}_{L^K}}(t)=\infty$ for a.e.
$t\in [0,\infty).$
\end{corollary}
\begin{proof}
By \cite[Section 60]{Gla66} condition \eqref{6.33} yields the
equality $\sigma_c(T)= \R_+,$ in particular $0\in \sigma_c(T)$ and
$t_1=0.$ Since $q\ge 0,$\ we have $0\le t_0 \le t_1=0,$ that is
$t_0 = t_1=0.$ It remains to apply Proposition \ref{prop6.12}
(ii)-(v).
\end{proof}
\begin{remark}
{\em
Condition \eqref{6.33} is satisfied whenever
$\lim_{|x|\to\infty}q(x) = 0.$ Thus, in this case the
conclusions of Corollary \ref{cor6.13} are valid.
However, it might happen that ${\sigma}(L^F) = {\sigma}_{ac}(L^K) =
{\sigma}(L^N) = [t_0,\infty)$, $t_0 > 0$ though $\inf q(x) = 0$.
}
\end{remark}
\begin{appendix}
\section{Appendix: Absolutely continuous closure}
Let us recall some basic facts of the $ac$-closure of a
Borel set of $\R$ introduced in\cite{BMN02}, see also \cite{GMZ08}.
\begin{definition}[\cite{BMN02}]
Let $\delta\in \mathcal \cB(\R)$. The set ${\rm cl}_{ac}(\delta)$
defined by
\begin{displaymath}
{\rm cl}_{ac}(\delta) := \{x \in \R: \; |(x-\varepsilon, x +
\varepsilon) \cap \delta| > 0 \;\; \forall \; \varepsilon
> 0\}.
\end{displaymath}
is called the absolutely continuous closure of the { Borel
set $\delta\in \mathcal B(\R)$.}
\end{definition}
Obviously, two Borel { sets $\delta_1, \delta_2\in \mathcal
B(\R)$ have the same $ac$-closure if their} symmetric difference
$\delta_1 \bigtriangleup \delta_2$ has Lebesgue measure zero.
Moreover, the set ${\rm cl}_{ac}(\delta)$ is always closed and
${\rm cl}_{ac}(\delta) \subseteq \overline{\delta}$. In particular, if
we have two measurable non-negative functions $\xi_1$ and $\xi_2$
which differ only on a set of Lebesgue measure zero, then
${\rm cl}_{ac}({\rm supp\,}(\xi_1)) = {\rm cl}_{ac}({\rm supp\,}(\xi_2))$.
\begin{lemma}\Label{III.6}
If $\delta\in \mathcal \cB(\R)$, then $|\delta \setminus
{\rm cl}_{ac}(\delta)| = 0$.
\end{lemma}
\begin{proof}
Since ${\rm cl}_{ac}({\delta})$ is closed the set ${\Delta} := \R \setminus
{\rm cl}_{ac}({\delta})$ is open. The open set ${\Delta}$ is decomposed as
${\Delta} = \bigcup^L_{l=1}{\Delta}_l, \ 1 \le L \le \infty$,
where ${\Delta}_l = (a_l,b_l)$ are disjoint open intervals. We set
${\Delta}_l = {\delta} \cap {\Delta}_l$, $l =1,2,\dots,L$. Obviously,
\begin{displaymath}
{\delta} \setminus {\rm cl}_{ac}({\delta}) = {\delta} \cap {\Delta} = \bigcup^L_{l=1}{\Delta}_l.
\end{displaymath}
We note that ${\Delta}_l \cap {\ | -0.528264 | -1.495464 | -0.09093 | 1.799233 | [
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320,
5030,
2802
] | 1,332,863 | 65,536 | ArXiv | -1 | -1 |
rm cl}_{ac}({\delta}) = \emptyset$, $l
=1,2,\ldots,L$. Hence for each $t \in
{\Delta}_l$ there is a sufficiently small neighborhood $\cO_t$ such
that $|\cO_t \cap {\delta}| = 0$. If $\eta$ is sufficiently small, then
$[a_l+\eta,a_l-\eta] \subseteq (a_l,b_l)$ and
$\{\cO_t\}_{t\in {\Delta}_l}$ performs a covering of
$[a_l+\eta,a_l-\eta]$. Since $[a_l+\eta,a_l-\eta]$ is compact we can
chosen a finite covering $\{\cO_{t_m}\}^M_{m=1}$ of
$[a_l+\eta,a_l-\eta]$. By $[a_l+\eta,a_l-\eta] \subseteq
\bigcup^M_{m=1}\cO_{t_m}$ we find $|[a_l+\eta,a_l-\eta] \cap {\delta}| =
0$ for each sufficiently small $\eta > 0$. Using that we get
\begin{eqnarray*}
\lefteqn{
|(a_l,b_l) \cap {\delta}| = |(a_l,a_l+\eta) \cap {\delta}| + |(b_l-\eta,b_l) \cap {\delta}| = }\\
& &
|(a_l,a_l+\eta) \cap {\delta}| + |(b_l-\eta,b_l) \cap {\delta}| \le 2\eta
\end{eqnarray*}
for sufficiently small $\eta >0$. Hence $|{\Delta}_l| = |(a_l,b_l) \cap
{\delta}| = 0$ which yields that $|{\delta} \setminus {\rm cl}_{ac}({\delta})| = 0$.
\end{proof}
\begin{lemma}\Label{III.7}
If $\{{\delta}_k\}_{k\in\N}$, ${\delta}_k \subseteq \R$, is a sequence of
Borel subsets, then
\begin{equation}\Label{3.16}
{\rm cl}_{ac}({\delta}) = \overline{\bigcup_{k\in\N}{\rm cl}_{ac}({\delta}_k)},
\quad {\delta} = \bigcup_{k\in\N}{\delta}_k.
\end{equation}
\end{lemma}
\begin{proof}
We set $\wh {\delta}_k = {\delta}_k \cap {\rm cl}_{ac}({\delta}_k)$ and ${\Delta}_k := {\delta}_k
\setminus {\rm cl}_{ac}({\delta}_k)$. We have
${\delta} = \wh {\delta} \cup {\Delta},$ $\mbox{where}\quad \wh {\delta} :=
\bigcup_{k\in\N}\wh {\delta}_k$ $ \mbox{and} \quad {\Delta} :=
\bigcup_{k\in\N}{\Delta}_k.$
By Lemma \ref{III.6}, $|{\Delta}_k| = 0$, $k \in \N$, which yields
$|{\Delta}| = 0$. Hence ${\rm cl}_{ac}({\delta}) = {\rm cl}_{ac}(\wh {\delta})$.
Similarly one gets ${\rm cl}_{ac}({\delta}_k) = {\rm cl}_{ac}(\wh {\delta}_k)$, $k \in
\N$. Notice that $\wh {\delta}_k \subseteq {\rm cl}_{ac}(\wh {\delta}_k)$, $k \in
\N$. We have
\begin{displaymath}
{\rm cl}_{ac}(\wh {\delta}) \supseteq \bigcup_{k\in\N} {\rm cl}_{ac}(\wh {\delta}_k) \supseteq
\bigcup_{k\in\N}\wh {\delta}_k = \wh {\delta}.
\end{displaymath}
Hence
\begin{displaymath}
{\rm cl}_{ac}(\wh {\delta}) =\overline{{\rm cl}_{ac}(\wh {\delta})} \supseteq
\overline{\bigcup_{k\in\N} {\rm cl}_{ac}(\wh {\delta}_k)}
\supseteq \overline{\wh {\delta}} \supseteq {\rm cl}_{ac}(\wh {\delta})
\end{displaymath}
which yields ${\ | -0.902186 | -1.792708 | -0.390048 | 1.419433 | [
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] | 1,332,863 | 66,560 | ArXiv | -1 | -1 |
The members of H.A.C.K. – one of the less wealthy hackerspaces – felt happy at first, when the place could afford to buy a slightly used UNI-T UT2025B digital storage oscilloscope. Besides being useful as a part of the infrastructure, having a USB and an RS-232 port seized our imagination – one of the interesting use-cases is the ability to capture screenshots from the device to illustrate documentation. As I tried interfacing the device, I found that supporting multiple platforms meant Windows XP and 2000 for the developers, which are not very common in the place.
I installed the original software in a virtual machine, and tried the serial port first, but found out, that although most of the functionality worked, taking screenshots is one available only using USB. I connected the scope using USB next, and although the vendor-product tuple was present in the list of USB IDs, so lsusb could identify it, no drivers in the kernel tried to take control of the device. So I started looking for USB sniffing software and found that on Linux, Wireshark is capable of doing just that. I forwarded the USB device into the VM and captured a screenshot transmission for analysis. Wireshark was very handy during analysis as well – just like in case of TCP/IP – so it was easy to spot the multi-kilobyte bulk transfer among tiny 64 byte long control packets.
I started looking for simple ways to reproduce the exact same conversation using free software – I've used libusb before while experimenting with V-USB on the Free USB JTAG interface project, but C requires compilation, and adding things like image processing makes the final product harder to use on other computers. For these purposes, I usually choose Python, and as it turned out, the PyUSB library makes it possible to access libusb 0.1, libusb 1.0 and OpenUSB through a single pythonic layer. Using this knowledge, it was pretty straightforward to modify their getting started example and replicate the "PC end" of the conversation. The core of the resulting code is the following.
Using this, I managed to get a binary dump of 38912 bytes, which contained the precious screenshot. From my experience with the original software, I already knew that the resolution is 320 by 240 pixels – which meant that 4 bits made up each pixel. Using this information, I started generating bitmaps from the binary dump in the hope of identifying some patterns visually as I already knew what was on the screen. The first results were the result of converting each 4-bit value to a pixel coloured on a linear scale from 0 = black to 15 = white, and looked like the following.
Most of the elements looked like they're in the right spot, and both horizontal and vertical lines seemed intact, apart from the corners. Also, the linear mapping resulted in an overly bright image, and as it seemed, the firmware was transmitting 4-bit (16 color) images, even though the device only had a monochrome LCD – and the Windows software downgraded the quality before displaying it on the PC on purpose. After some fiddling, I figured out that the pixels were transmitted in 16-bit words, and the order of the pixels inside these were 3, 4, 1, 2 ("mixed endian"). After I added code to compensate for this and created a more readable color mapping I finally had a script that could produce colorful PNGs out of the BLOBs, see below for an example.
In the end, my solution is not only free as in both senses and runs on more platforms, but can capture 8 times more colors than the original one. All code is published under MIT license, and further contributions are welcome both on the GitHub repository and the H.A.C.K. wiki page. I also gave a talk about the project in Hungarian, the video recording and the slides can be found on the bottom of the wiki page.
The Chaos Communication Camp was really great this year, and for those who were unable to attend (or just enjoyed the fresh air and presence of fellow hackers instead of sitting in the lecture room), the angels recorded and made all the talks available on the camp2011 page of CCC-TV.
Jayson E. Street gave a talk titled Steal Everything, Kill Everyone, Cause Total Financial Ruin! Or: How I walked in and misbehaved, and presented how he had entered various facilities with minimal effort and expertise, just by exploiting human stupidity, recklessness and incomp | 0.618154 | 1.466236 | 0.856356 | 0.63642 | [
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11407,
5199,
20239,
537,
29892,
25527,
2222,
2264,
322,
297,
2388
] | 1,332,869 | 0 | C4 | 12 | 5 |
etence. It's not really technical, and fun to watch, stuffed with photographical evidence and motivation slides.
While many hackers penetrate at high-level interfaces, Dan Kaminsky did it low level with his Black Ops of TCP/IP 2011 talk. Without sploilers, I can only mention some keywords: BitCoin anonimity and abuse, IP TTLs, net neutrality preservation, and the security of TCP sequence numbers. The combination of the technical content and his way of presenting it makes it worth watching.
Three talented hackers from the Metalab radio crew (Metafunk), Andreas Schreiner, Clemens Hopfer, and Patrick Strasser talked about Moonbounce Radio Communication, an experiment they did at the campsite with much success. Bouncing signals off the moon, which is ten times farther than communication satellites requires quite a bit of technical preparation, especially without expensive equipments.
I'll expand the lists as the angels upload more videos to the CCC-TV site.
Of course, life is never perfect, so there's one catch: it's not that well documented, there are not as many forum or blog posts and tutorials about CGA as with VGA or composite video. The pinout and the frequencies can be found on almost every ontopic web search result for the right keywords, the Wikipedia page has a pretty decent summary including colors, but most of the article deals with the PC-side hardware (video card), not the display nor the connection between them.
Multiplying the numbers in parentheses gives the exact length of each period, which makes it possible to write a simple sketch for an Arduino to display something simple. I chose to test with three horizontal displaying the flag of Hungary using a 66 row high light red (12), a 68 row high white (15) and a 66 row high light green (10) stripe. For the sake of simplicity, I connected the high intensity pin (6) to constant 5 volts, so the Arduino had 5 wires connected to it using the following scheme.
Those who used Arduino digitalWrite exclusively, might not know what PORT B is – if you're not one of them, you can skip this paragraph. The AVR microcontroller used in the Arduino has its I/O pins grouped into 8-bit registers that are mapped into the memory, thus accessible via certain variables, for example assigning an 8-bit value to PORTA writes the bits given to digital pins 0 to 7 in one quick step. In most cases, there's no need to get into this, but in timing-critical cases as this, there's significant advantage in accessing the hardware directly – see for yourself in Bill Grundmann's thorough blog post. You can read more about this direct access on the official Arduino page about port registers.
As you can see, I arranged all five pins to be connected to the low five bits of PORT B, which means that I can modify all their values in a single instruction. At the beginning of my sketch, I used #defines to provide constants named in a meaningful way.
In the setup function, the sketch initializes the output ports and sets all output pins to low.
There are also two global variables used to track the color of the current stripe (rgb) and the number of the current row (row).
In the loop function, the sketch draws a single row. It begins with the left blanking and overscan area that takes around 6.7 μs, then sets the R-G-B pins to the color of the current stripe. The width of the visible area is 640 pixels that take approximately 44.69 μs, after that, all R-G-B pins are reset to low.
The right overscan and blanking take around 7.8 μs, after that, HSYNC needs to be pulled high for approximately 4.47 μs.
At the end of the row, the row-level logic increments the row counter and uses a switch statement to handle certain rows specially. The first three cases cover the flag generation: after the 66th row, the color changes to white, at the 134th it does again to green, and at the bottom of the screen, color gets turned off. Between the 225th and the 228th row, VSYNC is set to high, and after the last row, the row counter gets reset to zero.
The actual delay values seen in the snippets above are the results of rounding and testing as | 0.033645 | 1.229873 | 0.889327 | 0.644224 | [
0.10041680777739512,
-0.033126546549984016
] | [
1.3480545078773314,
1.1116915351444485
] | [
0.7346167104547956,
1.0440381155516765
] | [
0.23320794369381947,
1.0552409484617309
] | 1,024 | [
512,
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] | 1,332,869 | 1,024 | C4 | 12 | 5 |
Q: How to draw and format this image (equilateral triangles) in TikZ? I'm writing up a problem set and thought it would be cool to figure out how to do this in TikZ for a nice look rather than just insert the image. I would also like to format it in the center of my page. If anyone could help, that would be great!
A: No need to call the calc library for this, just put some nice coordinates while you draw your triangles.
\documentclass{article}
\usepackage{tikz}
\def\r{3}
\begin{document}
\begin{tikzpicture}
\draw (0,0) --++ (0:\r) coordinate[pos=1/3] (a) coordinate[pos=2/3] (b) --++ (120:\r) coordinate[pos=1/3] (c) -- cycle coordinate[pos=2/3] (e);
\draw (e) -- (c) coordinate[pos=0.5] (d) -- (b) -- (d) -- (a) -- cycle;
\end{tikzpicture}
\qquad
\begin{tikzpicture}
\draw (0,0) --++ (0:\r) coordinate[pos=0.5] (a) --++ (120:\r) coordinate[pos=0.5] (b) -- cycle coordinate[pos=0.5] (c);
\draw (a) -- (b) coordinate[pos=0.5] (d) -- (c) coordinate[pos=0.5] (e) -- cycle coordinate[pos=0.5] (f);
\draw (d) -- (e) -- (f) -- cycle;
\end{tikzpicture}
\qquad
\begin{tikzpicture}
\draw (0,0) --++ (0:\r) coordinate[pos=0.25] (a) coordinate[pos=0.5] (b) coordinate[pos=0.75] (c) --++ (120:\r) coordinate[pos=0.25] (d) -- cycle coordinate[pos=0.75] (g);
\draw (g) -- (d) coordinate[pos=1/3] (f) coordinate[pos=2/3] (e) -- (c) -- (e) -- (b) -- (f) -- (a) -- cycle;
\end{tikzpicture}
\end{document}
A: Here is a part of the code that uses the Tikz calc library.
I'll let you look for the other figures.
PS: it would have been desirable that you show that you searched before asking the question.
\documentclass[tikz]{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\usepackage{graphicx}
\begin{document}
\begin{tikzpicture}
\draw (0,0) coordinate(A) --++(60:5cm) coordinate(B)--++(-60:5cm) coordinate(C)-- cycle;
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-- ($(A)!1/3!(C)$) -- (AB);
\end{tikzpicture}
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A: A pstricks solution – more specifically, with pst-eucl (which mimicks geometric transformations) for the first figure. The second and third figures can be done similarly.
\documentclass{article}
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How the U.S. Can Survive Its World Cup Group of Death
Omam-Biyik in 1990. Why not be the jumper rather than the jumpee?
Taking their cue from Jürgen Klinsmann's evaluation of "the worst of the worst" for the American schedule, fans are assuming the U.S. will get ushered out of the 2014 World Cup in Brazil without winning or drawing or maybe even scoring.
Having covered the past eight World Cups, I would like to note that the matches have yet to be played – and that group play is always full of revelations.
Why not be one of those teams that upset the favorite? Why not make your own luck? Why not catch players before they are acclimated? Why not be better than anybody expected? It has happened. Every four years it has happened.
The first thing to remember about the World Cup is that the best teams come into the tournament with the most players chewed up by the insatiable club schedules, including the Champions League. Outsiders can have fresh legs and fresh attitudes – very often worth a victory in the first round.
Without even discussing the American team here, let's look at a few weird and wonderful things that have happened in the first match of group play:
In Spain in 1982, Algeria stunned West Germany, 2-1 – still one of the great upsets in World Cup history. The West Germans used a disgraceful 1-0 waltz victory over their friends from Austria in the third match, so both teams could advance past Algeria. FIFA later tightened the rules against manipulation like that, but Algeria will always have the pride of that upset.
In Mexico in 1986, in the opening match in Azteca, Bulgaria tied the defending champions of Italy, 1-1, in the 85th minute. Then again, Italy almost always has a wretched first round – part of its charm.
In the very first match in Italy in 1990, Francois Omam-Biyik of Cameroon outjumped Roberto Sensini of Argentina for a header goal in the 67th minute for a 1-0 victory. (The Argentina coach yanked Sensini two minutes later.) Was this a huge upset? Not really. Ancient Roger Milla helped Cameroon become the first African team to reach the quarterfinals, and Argentina reached the finals.
In 1994, Ireland beat Italy, 1-0, in Giants Stadium in the first match. Once again, Italy had to stagger into the knockout round, eventually losing the final when two injured stars missed in the shootout.
In 1998, in France, talented Spain sauntered into a first match against Nigeria, and was out-run, 3-2, by Bora Milutinovic's team. Spain did not reach the knockout round, its glory still a generation away.
In 2002 in South Korea, Zinedine Zidane pulled a thigh muscle in an exhibition against the aggressive Reds, and the lethargic defending champion, France, was stunned by its former colony, Senegal, 1-0, in the opening match, and did not survive the first round.
In 2006, Trinidad & Tobago, in its long-delayed World Cup debut, held Sweden to a 0-0 draw in the first round in Germany. When bonuses did not materialize, the Soca Warriors did not score in two straight losses.
In 2010, France, England and Italy all staggered to draws in their first matches in South Africa, and only England made it into the knockout round. All three were exposed as weary, disinterested or, in France's case, mutinous.
From my eight World Cups, I have learned that national teams are all-star aggregations, thrown together a few weeks before the opening match. Very often, African teams have talent but struggle with financial and logistical issues. The U.S. cannot think about the losses to Ghana in 2006 and 2010 and must get 3 points from the first match. Then it will have to hold Cristiano Ronaldo and Portugal to a draw, somehow. After that comes Germany, always talented and never nonchalant.
The U.S. will have to hope the other teams chew each other up in the first round. In the most recent | 0.76197 | 1.376705 | 0.238782 | -0.08481 | [
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] | 1,332,880 | 0 | CommonCrawl | 5 | 7 |
World Cup, four teams advanced with 4 points.
(http://www.fifa.com/worldcup/archive/southafrica2010/standings.)
The great thing about having a former champion like Klinsmann as coach is that he can, without boring his players, remind them of disasters and upsets he has seen in the World Cup. The first match is vital. The schedule:
http://www.fifa.com/worldcup/matches/index.html
Now if the U.S. can arrange a transplant of its back line.
Diop outruns Petit in the opener in Seoul, 2002.
Thor A. Larsen
With the overall improvement in recent years with US soccer and US players, specifically, I am optimistic of a very good showing of the American team. As noted in recents blogs on sports, team psychology and coaches are very important ingredients for winning and I believe this coach will be provide that added that intangble asset. My most memorable experience watching World Cup finals was in 1982 when Italy beat Germany. My Italian heritage wife was so excited as she never watched soccer before and we were with my German-American friend and wife who were ardent Germany fans. Point being that unlike any other sporting matches outside the Olympics, World Cup tournament stimulates the most ardent national passions. I really look forward to it and that US does very well. It will be easy to root for the US, as Norway did not make the cut!
John Wilcockson
... and don't forget, George, that USA beat England 1-0 in group play the last time the World Cup was held in Brazil in 1950. As for the busy Euro schedules, I think they will help the U.S. team with players such as Tim Howard—what a great game he played last Sunday against Arsenal!
John, I am honored to hear from the great cycling writer. We did not cover the 1950 World Cup, did we? I got to meet Harry Keough, the defender, in St. Louis a few years ago, very sweet guy. The U.S. papers did not believe the score, and attributed the victory to Your Lads. Good point -- anything can happen. Best, GV
Hooray! Nice intro to book--looking forward to it. Soccer Rules!
(no mention of Dolan, Knicks, Nets. Nirvana.).
Roy Edelsack
And, since GV is too much of a gentleman to mention it himself, a nice segue into mentioning that "Eight World Cups" is available for pre-order on Amazon (Kindle version too!).
John McDermott
While on paper, and as of today, it looks like the US has a terribly difficult task in front of it, I, like you, know all too well that what we think today can end up being very different from what happens when the games are actually played. The biggest factor that can alter possibilities is that of fitness. What if Ronaldo suffers an injury for Real Madrid that weakens him or keeps him out of games in the first round? Suddenly Portugal look a lot more beatable. What if the US arrives in Brazil with, say, Jozy Altidore on a scoring tear and Landon Donovan motivated and in great shape? What if Jürgen finds the right combination of four defenders to put in front of Howard(or Guzan, why not?) and they prove to be strong in defense, finally gelling as a unit when it really counts? All possible. And forget what happened in the past with Ghana(I'm sure Jürgen will be hammering home the point of the irrelevance of old results, apart from their usefulness as a learning tool). Ghana are eminently beatable(at least that's the belief that Jürgen will be trying to instill in his charges, and it's also probably quite true). I've heard two prominent former German national team stars-Michael Ballack and Bodo Ilgner-say that the USA game will be very difficult for Germany, in large part because of Jürgen. I think Ballack and Ilgner probably know what they are talking about. Germany are a very, very good team. But they are beatable too. In my view, there is a range of possibilities ahead for the US team, from three-and-out to playing with style and winning the group. That's why they play the games-to find out who really is better on that day. At this point only a fool would be "sure" of | 0.142877 | -0.010137 | -0.388938 | -0.819859 | [
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what's going to happen next summer in Brazil. I'm certainly not.
Andrew Tansey
My most profound lack of insight is that USA are not the only side drawn into the Group of Death. I have not taken the time to research it, but what are Germany, Ghana and Portugal saying about THEIR draws?
Aside from the conspiracy theories - not only the competition but also the travel schedule - I think attention the draw has gotten in the USA is a good sign about our advancement as a football nation. The Gloom of Doom resembles that ever present - without there ever having to be a particular reason for such GoD - in one of the other two English speaking teams in the tournament, England, of course. By the way, at The Public Theater on Monday, thanks to my 1/3 (only one generation to work with so far), Men In Blazers spoke of a clear conspiracy against all three English speaking teams, to get Australia into the mix. Not sure when that podcast will be available, but they seem to be more popular than Messi in the 20-something generation of soccer fans. http://meninblazers.com/
How can we express well enough our respect, admiration, gratitude and fondness for Sepp Blatter?
Andy Kusi-Appiah
"Ghana are eminently beatable(at least that's the belief that Jürgen will be trying to instill in his charges, and it's also probably quite true)."
The above is very true...what is also true is that the USA beatable, and for that matter all the 32 nations participating in the world cup - The ball is round indeed.
The most important thing is who "appears' to play on that day.
I wish both teams the very best in this world cup of death.
Nigeria, Ghana, Cameroon (just to name only a few) are English speaking teams in the world cup as well.
There are more than 3 English speaking teams int he world cup.
If there were not a Sepp Blatter, we would have to invent one.,
Alan Rubin link
It is refreshing to see a discussion on the US World Cup draw that is not all "doom and gloom", with many excellent reasons for otimism being expressed.
John McDermott correctly addressed the potential of the USMNT getting through the draw. It is a positive that the round begins with Ghana and finishes with Germany.
The team's talent pool is probably the deepest it has every been and with a collectively high level of experience.
The book, "The Numbers Game: Why Everything You Know About Socceris Wrong" was previously referred to in the Knicks-Nets topic. Through out the book authors David Sally and Chris Anderson relate statistics to the randomness of soccer. They authors concluded that soccer was a gome of 50% skill and 50% luck. Any game can suddenly turn on a lucky break. Teams being dominated for 89 minutes can win on their only shot of the game. This is an interesting book to read as they debunk many of soccer's cliches.
A major criticism of the draw is the excessive and unequal travel requirmentas. Also, the top eight teams should each be seeded into the one of the eight groups. The rankings should be based upon current perormance, or at least should not go back too far.
My favorite George Vecsey quote sums up the unpredictability of World Cup soccer, "I don't know any other sport so beloved by the people who play it, or the people who watch it.
Everybody in the stadium understands that at any given moment something amazing can happen, out of nothing."
I expect the US to surprise the experts.
Alan, If Lehigh could beat Muhlenberg 2-0 with you in goal then anything can happen!
Ed--Thanks for your kind show of confidence.
I can assure you that am not on Klinsmann's short goalie list, or any of his lists. However, that is only because of my age.
Brian Savin
I've only seen one game and part of another, but I see a difference of confidence. This team at least seems to play as if it expects to win. What the refs do to them once the competition starts in earnest i'd rather not think about and hope for the best.
Charlie Accetta link
GV - I love it that the USA is regularly qualifying to participate in the Big Dance but what I'm reading here, from you and the comment contributors, | 0.12918 | 0.12416 | -0.050984 | -0.750151 | [
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Baby, I love you so much. You look great in that negligee that I got you on sale from Sears when I was waiting for my car to be serviced. Darn alternator is always acting up. You know, you've driven it. I think that Carl, the guy at the counter had it out for me because he gave me a look of hatred when I brought it the car...kind of like the look you're giving me now. Baby, what's up? I just wanted to tell you about my day before we...you know...rocked the sheets. You don't want to hear about the Pinto? Strange, anyhow, I thought that we could have some fun before we got down to business and I picked this up. You know how you're always thirsty after we're done? Oh wait; maybe that's me, anyhow. Cleaning up chocolate is a mess. It doesn't come out of anything. The last thing I want you to do is worry about cleaning at a time like this. Also, I know you like strawberries, but it's not really in season so I did the next best thing. I stopped at a gas station on the way home from work today and picked up a can of Crave strawberry chocolate milk that I thought we could share while we're, you know, doing it. I know, strawberry and chocolate. You know how you've had your eyes on a Prius, but this is like a delicious hybrid of its own.
Drink up, baby. I keep calling you baby to try and get you in the mood because you don't really look like you're in to it anymore. I don't really get it but give this a quick swig. It's not bad. It tastes like chocolate covered gummy strawberries. That's cool, right? Plus, it comes in this pretty awesome aluminum can so it keeps cold longer. Why are you turning the TV on, I thought we were going to...oh turn that back! I love Psych. Move over, you're taking up the whole bed.
Eggnog is one of the foulest beverages in the history of the world. Eggs and milk mixed? No thanks! Ever since I was a kid I loathed the stuff. I don't give a rip if Santa loves the stuff. You know what I do care if he likes it. He can have my share.
A couple of years back a friend got a carton of Silk Nog and I went off about how gross the stuff was. I was then basically forced to try it. I was shocked that I actually enjoyed the stuff. I don't know if it was just that I knew it wasn't made out of disgusting ingredients or if it's just better than eggnog. I will tell you this, it's not as thick as eggnog, but it's by no means watery. When you go to take a sip and breath in through your mouth, a strong taste hits you before the liquid. It's like an aftertaste in reverse. I think Silk has mastered quantum physics with this beverage. I've never had that happen before with a drink. When it finally hits your tongue it's very creamy, and tastes like light holiday spices.
Every since that day I've gotten a handful of cartons of this a year. It's not an all the time drink, but a nice glass of it every once and a while is really nice. I guess I have to get in the holiday mood in some way, and I choose to do so eating cookies and drinking Silk Nog.
Ooooh. So let me get this straight. You're vegan and it's Christmas? Well that is very saddening. You can't have any milks or meats. That's unfortunate. And now you're telling me that you used to love Hot Chocolate with milk like mama used to make? Oh, and you used to put a peppermint stick in it to really hit it home? I'm very sorry that you can no longer have that. Let me present to you, though, something very on par that is not just for you, but also for the rest of us. It's from West Soy and it's chocolate with peppermint stick and tastes just like (cold) hot chocolate with mama's recommendation of puttin' a peppermint stick in it. What more could you ask for? W | -0.38646 | -1.817777 | -1.837132 | -2.217283 | [
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Sergei Prokofiev wrote the Overture on Hebrew Themes, Op. 34, in 1919 while he was in the United States. It is scored for the rare combination of clarinet, string quartet and piano. Fifteen years later the composer prepared a version for chamber orchestra, his "Op. 34 bis" or Op. 34a, retaining a separate part for piano but featuring solo cello as much as solo clarinet.
Until recently the chamber orchestra version had been the better known, being easily programmable by orchestras, while the original version (for six instruments) requires out-of-the-way planning on the part of a string quartet. But two recordings have drawn attention to the original: Michel Portal clarinet, Parrenin Quartet, Michel Béroff piano (1974); and Giora Feidman clarinet, Juilliard String Quartet, Yefim Bronfman piano (1994).
Background
Prokofiev arrived in New York in September 1918. Early in 1919, he was commissioned by a Russian sextet called the Zimro Ensemble, which had just arrived in America from the Far East on a world tour sponsored by the Russian Zionist Organization. The members played the instruments in this work's instrumentation, and were led by their clarinetist Simeon Bellison, who was trained in Moscow and had been principal clarinettist of the Mariinsky Theatre from 1915. (Bellison would soon become principal clarinet with the New York Philharmonic Symphony Orchestra.) They gave Prokofiev a notebook of Jewish folksongs, though the melodies Prokofiev chose have never been traced to any authentic sources. It has been suggested that Bellison had actually composed them himself in the Jewish style . The other members of the ensemble were Jacob Mestechkin, first violin, G. Besrody, second violin, K. Moldavan, viola, Joseph Cherniavsky, cello, and Leo Berdichevsky, piano , although one source names the second violinist as Michael Rosenker, who was 19 years old at the time .
It received its premiere at the Bohemian Club in New York, on 2 February 1920, with Prokofiev as guest pianist. Before they disbanded in 1922, the Zimro Ensemble performed it again at least twice at Carnegie Hall—with their own pianist, Berdichevsky, in 1921, and possibly with guest pianist Lara Cherniavsky in December 1920 . Prokofiev regarded the work as conceived essentially for a sextet, and long resisted suggestions of arranging it for other forces. In September 1930, he remarked, "I don't understand what sort of obtuse people could have found it necessary to reorchestrate it" but, nevertheless, he was persuaded to make a version for chamber orchestra in 1934, published as Op. 34a . The orchestrated version is performed less often, because it lacks the earthiness and uncanny timbral balance of the original .
Prokofiev did not regard the work very highly. When a Scots critic, Andrew Fraser, published an article in 1929 describing the Overture as "a beautiful and pathetic work", the composer wrote in response, "its technique is conventional, its form is bad (4 + 4 + 4 + 4)". When his friend Nikolai Myaskovsky praised the second theme, Prokofiev retorted, with reference to the work's coda, "from the musical point of view, the only worthwhile part, if you please, is the final section, and that, I think, is probably the result of my sweetness and diatonicism" .
Analysis
Its structure follows the form of a fairly conventional sonata form. It is in the key of C minor. The clarinet and the cello are prominent, introducing the first and second themes respectively. However, all instruments are balanced well, and each instrument plays both themes, often in imitation. Zimro's pianist, Leo Berdichevsky, was a graduate of the Petrograd and Berlin conservatories .
The first theme, Un poco allegro, has a jumpy and festive rhythm, unmistakably evoking klezmer music by alternating low and high | 1.701567 | 1.394611 | 0.727008 | 0.961592 | [
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1.6303442619690105
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Services / facilities within the grounds include rugby pitches, football pitches, cricket pitches, changing pavilion, a cafe, playground, outdoor gym, gardens and a refreshment kiosk.
The park is generally opened from 8am until dusk. .
The toilet block has reduced opening hours due to anti-social activity.
They are generally open from at least 9:00 - 15:00.
The cafe and refreshment kiosk opening hours also vary accoring to the season and weather.
Both facilities are open every day of the week.
The team changing pavilion is open at weekends or when matches are playing.
The pavilion was closed for refurbishment at the time of our survey (February 2015).
The nearest National Rail station is Charlton.
The dimensions of the designated parking bay(s) are 440cm x 560cm (14ft 5in x 18ft 4in).
The nearest designated bay is 20m (21yd 2ft) from the access point into the park at the end of the car park.
The furthest designated bay is 30m (32yd 2ft) from the access point into the park at the end of the car park.
This car park is located at the eastern end of the park, off Charlton Park Lane.
It is close to a school and hospital and is often full on weekdays.
There is direct access from the car park onto footpaths into the park.
The car park is located near to the changing pavilion and playground.
There is a public car park off Torrance Close, which is off Charlton Park Road.
There is also some on-street parking in Cemetery Lane, Canberra Road and Charlton Park Lane.
You are on Outside Access (Charlton Park Road Entrance). Skip to the next section.
This information is for the entrance located on Charlton Park Road, opposite Charlton Lane.
The door opening is 350cm (11ft 6in) wide.
You are on Level Change (Charlton Park Road Entrance). Skip to the next section.
You are on Outside Access (Canberra Road Entrance). Skip to the next section.
This information is for the entrance located on Canberra Road.
You are on Level Change (Canberra Road Entrance). Skip to the next section.
You are on Outside Access (Cemetery Lane Entrance). Skip to the next section.
This information is for the entrance located on Cemetery Lane.
There are wide, easy to moderate tarmac paths throughout the park.
Near the Charlton Park Road entrance is an area with outdoor gym equipment and two table tennis tables.
You are on Playground. Skip to the next section.
There is an enclosed playground at the eastern end of the park, near the car park.
There are heavy outwards opening gates 130cm wide.
The playground is generally level with some uneven areas.
Play equipment which might be suitable for disabled children includes a level access roundabout.
There are benches and picnic benches in the playground. Most are located on grassed areas.
You are on Restaurant Bar Internal (Old Cottage Coffee Shop). Skip to the next section.
The café is located close to the Charlton Park Road entrance and the toilet block.
There is level access with a small raised threshold at the entrance.
The double doors are held open.
In fine weather, there is additional seating with arm rests outside the venue.
There is also a refreshments kiosk at the changing pavilion next to the playground (picture 5).
There is a medium unmarked step up to the kiosk.
Opening times vary according to the season and weather.
You are on Outside Access (Toilet Block Entrances). Skip to the next section.
This information is for the entrance located at the male and female toilets.
This accessible toilet is approximately 20m (21yd 2ft) from the Charlton Park Road entrance.
This accessible toilet is located within the female toilet block.
This is a female facility.
The dimensions of the accessible toilet are 225cm x 250cm (7ft 5in x 8ft 2in).
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Posted on October 1, 2016 by Pomeroy Daily Sentinel
Buckeyes blast Meigs, 47-6
By Bryan Walters - [email protected]
NELSONVILLE, Ohio — Host Nelsonville-York scored the first 41 points of regulation and never looked back Friday night during a resounding 47-6 victory over the Meigs football team in Week 6 Tri-Valley Conference Ohio Division contest at Boston Field in Athens County.
The visiting Marauders (3-3, 1-2 TVC Ohio) were simply overmatched by the Buckeyes, who scored 14 points in each of the first two periods en route to a commanding 28-0 halftime cushion.
NYHS (4-2, 2-0) tacked on 13 more points in the third canto as the Orange and Brown extended their lead out to 41-0, but the Meigs finally cracked its way into the scoring column late in the fourth as Christian Mattox plunged in on a one-yard run — making it a 41-6 contest with five minutes left in regulation.
Nelsonville-York concluded the scoring with 2:51 remaining in the game as Keegan Wilburn scampered 50 yards to paydirt while wrapping up the 41-point outcome.
Garrett Maiden started the Buckeye scoring at the 7:17 mark of the opening canto after hauling in a 13-yard pass from Hunter Edwards, making it a 7-0 contest.
Alex Mount followed with a 16-yard touchdown run with just 12 seconds remaining in the first for a 14-0 edge through 12 minutes of play.
Mount added scoring runs of 42 and 21 yards in the second period, allowing NYHS to double its lead to 28-0 with 6:58 remaining in the half.
Aron Davis gave the Buckeyes a 35-point edge after hauling in a 31-yard pass from Edwards at the 9:38 mark of the third, then Wilburn broke free on a 25-yard scoring run at the 4:30 mark of the third for a 42-0 advantage.
Nelsonville-York outgained the Maroon and Gold by a sizable 409-111 overall mark in total yards, which included a 347-99 advantage on the ground. The hosts claimed an 18-7 edge in first downs and were penalized six times for 88 yards, compared to only three penalties for 28 yards on the Marauders.
NYHS — which finished the night plus-1 in turnover differential — averaged 9.9 yards per carry on the evening after producing 347 rushing yards on 35 attempts. Meigs, conversely, had 99 yards on 35 attempts for an average of 2.8 yards per tote.
Lane Cullums led MHS with 37 yards on 13 rushes, followed by Christian Mattox with 33 yards on 11 attempts. Cole Adams also had three carries for 30 yards.
Zach Helton finished the night 5-of-8 passing for three yards, while Mattox went 1-for-4 for nine yards. Neither Meigs quarterback threw an interception. T.J. Williams had nine receiving yards on a single catch to lead MHS.
Mount led the Buckeyes with 134 rushing yards on nine carries, followed by Wilburn with 106 yards on 11 totes. Edwards added 58 rushing yards on seven carries and also completed 5-of-12 passes for 62 yards.
Maiden hauled in a team-high three passes for 25 yards and Wilburn also had two grabs for 11 yards.
The Marauders — who are now 3-0 in odd weeks and 0-3 in even weeks this fall — return to action Friday when they host Athens in a Week 7 TVC Ohio contest at Farmers Bank Stadium/Holzer Field. Kickoff is scheduled for 7:30 p.m.
Bryan Walters can be reached at 740-446-2342, ext. 2101.
By Bryan Walters
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Bush family rallying around Cruz?
Billionaire Donald Trump has electrified millions of frustrated Republican voters and stormed through the GOP primary with a commanding delegate lead.
His success in the 2016 presidential nomination race, according to political analysts, has revealed a deep-seeded anger among voters toward establishment Republicans… and it has party officials so anxious, they're pulling out all the stops to stop it.
Now, rumors indicate that the Bush political dynasty may be set to weigh in. Former Florida Gov. Jeb Bush may no longer be onstage or in the race, but the onetime presumed front-runner is involved again. Bush has summoned Texas Sen. Ted Cruz, as well candidates John Kasich and Marco Rubio — but not Trump — for private meetings ahead of Thursday night's GOP debate.
Cruz is seeking Bush's backing before the critical winner-take-all Florida primary on Tuesday.
Meanwhile, Bush's brother, Neil, has already signed on with Cruz's campaign as a member of his finance team.
There is no love lost between the Bushs and Trump. Jeb and The Donald engaged in heated confrontations throughout the campaign, repeatedly referring to each other as "loser."
And the family wouldn't be the first within the party establishment to realize that while Cruz may not be their perfect candidate, he may be the GOP's best last chance to stop Trump.
Other possible Cruz supporters include reluctant Senate colleagues and former presidential rivals with strong ties to major donors, who have long feared Cruz's purist ideology but dread the prospect of a Trump nomination even more. The first-term Texas senator on Wednesday announced the backing of one former primary opponent, Carly Fiorina.
"It's an outsider year, and the most logical person to take on Trump based on past performance is Ted Cruz," said another former presidential opponent, South Carolina Sen. Lindsey Graham. Earlier in the year, Graham likened the choice between Cruz and Trump to "being shot or poisoned."
"He's not my preference," Graham said of Cruz. "But we are where we are. And if Trump wins Florida and Ohio, I don't know if we can stop him."
Indeed, next Tuesday's winner-take-all contests in Florida and Ohio have injected a sense of urgency into the GOP's anti-Trump movement.
"I could see myself as a Cruz supporter," Nevada Sen. Dean Heller, a Rubio supporter, said Wednesday. "I agree with Cruz 80 percent of the time. There are things I don't agree with him on, but if we are agreeing 80 percent of the time and he's a conservative, we are going to get along just fine."
"The Republican establishment is in its death throes," said longtime tea party leader Mark Meckler. "The only remaining candidates are 100 percent anti-establishment."
How hypocritical of the Republican Party. Makes me ashamed to say I am a Republican and might change to Libertarian or Independent. Listen to the people not your special interests. We the people in the majority want Trump and there are more, once they stop feeling guilty about betraying their chosen candidate and see through all the manipulation and lies will also vote for Trump.
J.P. says
Crews is still part of the establishment and we do not need another Bush in the White House. Think about it People long and hard before you make that last commitment in Nov..
Allen says
You apparently don't know Ted Cruz if you think in any way he could be called a member of the establishment. Trump and Cruz are leading this race precisely because they are NOT members of the establishment. No one else has even gotten close.
Cruz has always been with Bush. Cruz wife worked with Bush. Jeb Bush, Rubio and Ted Cruz all grew up together in the political life. People are in denial if they think Cruz and Rubio are not part of the Bush Dynasty and if the Bush Dynasty can not have one of their own named Bush running they will support one of the other two. Only non establishment politician running is Donald Trump. Vote Donald Trump and for get Cruz, Rubio and Kasich.
Shelton says
Just shows me how ill-informed you are. The Republican establishment hates Cruz because he has fought them his entire time in office. Cruz wants to protect, and has proven it, our constitutional rights which | -0.025593 | -0.55807 | -0.943741 | -0.412582 | [
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are under attack by both parties and particularly Obama. If people with your views don't wake up you will be wearing a burka and won't be able to even drive a vehicle. Trump is a liberal in disguise and will deceive the American people just like Obama has done. Wake up America.
Ferdinand Holcapek says
I agree with you, but if I have to choose between Trump or Cruz then Cruz is more aligned with my thinking then Trump. I hate egotistical, narcistic bullies
JCS says
I totally agree with Shelton
sambad says
How could you be so dishonest as to even put comments like that in print? Cruz is an outsider, a very honest man and he will use the constitution as his road map in leading our country. Trump is the one who constantly lies and flip flops. He said during his last debate that he would be flexible on immigration. An ego driven bankrupt who will twist and turn anything and everything to mislead our country is who you are supporting.
Mr. Cruz has fought for what was right. He made them mad because he stood up for what was right. The fact that he worked for Bush and they know his work ethics. If you want a hard for you. Vote for Cruz. He is not a "Teddy Bear" in the senate now. He will not be as a president. He is the president that will call a conference just to le you and the rest of the world what is working and what I'd not. The reason. What he will do to fix it.
Guy gascard says
Really ??????
Are you insane ?
Just because his wife worked with Bush means nothing Cruz has been one of the few in the Senate to stand up AGAINST the Establisment and for our Constitutional rights
bill b says
birds of a feather fly together, if i was running i'd dread a jeb Bush endoresment, hes a fool and a half. as far as the Bush family ( READ MY LIPS ) their irrelavant like all of the dinosaur freeloading politicos in the GOP will be the day after Trump takes the oath TRUMP in 2016
Sarah Abrams says
Donald Trump has paid off hundreds of companies/people, including Hillarity Clinton, to get bills passed by Democrat Lobbyists and Congress. Just because Trump gets the people who are "mad" at the country does not make him viable for Prez – he was/is even anti-Tea Party and all he knows is "the Art of the Deal". He has NO military background (just like BHO), no foreign country knowledge (like BHO – NATO, UN (other than lying about Benghazi)…he just knows how to quote the Kuran, kneel and say Islamic prayers, and he wants to shut down GITMO, so that he cannot be sent there when a new President sits in the White House.) Trump just garners the "frenzy", which is why he has the following that he does have. We will be very sorry IF he is sworn in UNLESS he chooses Dr. Ben Carson for his VP. I have already told Trump, Cruz and Rubio that I will NOT vote for them unless they choose Ben Carson for their running mate. You all should send that provisional note to each of the last 3.
When you endorse Cruz and his Bushes, Jeb and Neil of the Savings & Loan debacle( who the Bushes have not mentioned since, not even mentioning that he is one of the sons, because they know how that hurts their image as much as GW ever did) GW was a war criminal and Neil just destroyed peoples finances) you are just choosing the foxes to protect the henhouse! I think that the bushes were behind mittens trying to dictate who republicans should vote for!
Cruz is a typical lawyer with no morals and lies through his teeth. We really need a smart business man to finally run our beautiful country. Politicians do not have a clue what they are doing
Talking about a person that is a liar and dishonest is Trump and racist. Listen to his speeches everything is about him, how great, rich, and everyone loves him. It is not about the people rights is about him and power and control. Entertaining says what people want to hear.
You are supporting the only liar I've seen in any of the debates. It's amazing how so many of you Trumpsters can't see how much of | -1.624005 | -1.896564 | -2.146553 | -1.226811 | [
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a lying ego maniac you are following.
Philip says
Oh He/They know what they are doing alright.
And it isn't good, the last three elections has been nothing but LIES, and sorry excuses about what they can't do because of the other party. They are all LIARS, They lie, and promise what they are going to do.
Until they get back in office, and they have all the excuses why they can't do what they promised they would do. Need to get rid of the whole bunch of the LIARS and get some new blood in politics, and have term limits.
Jim man says
From the beginning business men have been failures as Presidents. Trump would be even worst as he is bad at business. He has had 38 failed businesses. 4 bankruptes. Still owes 1000's of venders. over $60 million dollars
1000's. Of people out of work Trump is not a good man. Does any one study or read.??
I think you are wrong about that Allen. Cruz talks a good show but I believe when it comes right down to it — he is owned by the traitorous establishment and will do whatever they tell him to do. I originally thought that Trump & Cruz would make a good team but I don't think that anymore. We don't need another career politician as our supposed leaders we need business people and patriots not traitors, scheming hypocrites.
kenneth cameron says
VOTE FOR TRUMP NOV. 3 2016 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Who in the frigg is the establishment. People keep pulling that term out of thier ass & don't have a clue what they are talking about I will write this real slow so you can understand There is no such thing as establishment.
Any one that thinks Trump is one phony dumb ass Is the establishment. I WANT SOME ONE THAT HAS EXPERIENCE. That have studied these issues for thousands of hours Or many years. Has debated them for hours on end Like any profession it takes years to get good at that profession I don't want some one that has not studied the issues Trump is to mentally lazy. And does not know the issues. Or how our government works
The President is not a dictatorship. We have 3 co equal branches of government They must work with each other
Or nothing gets done Trump has called them idiots & stupid at least 12 in the Republician senaters have said they will never work with him. There are 54 Republicians. 46 democratic- Other words he will never get a bill passed
He will be completely neutered You can not put taxes or tariffs on goods coming from Mexico or China Unless congress approves Other words. He can not do anything he says The house controls the purse strings. He can not spend money unless congress allocates money for said project Trump would be a complete failure
TRUMP IN 2016 !!!!! end of story.
joe german says
If you don't think so, check into Cruz's wife. She is part of the establishment. It's very scary what she is a part of.
Shirley messier says
We need TRUMP I hope the people give him a chance.
debdell says
You better take a look at the youtube video. Cruz fooled me in the beginning until I did the research. He is even lying about his name – it is not Ted. Maybe his real last name is Castro! Anyway Cruz is total establishment and so is his wife.
In 2010 Cruz and Goldman Sachs got together and planned a run for the presidency. Cruz copied Obama's game plan – be a senator for a year and then run for president. Cruz determined that he needed recognition and decided to act like he was against the establishment (Goldman Sachs and others) while making a name for himself. He got nasty with other senators calling them liars etc. which worked and got him known as going against the establishment. To get elected as senator in 2012, he "borrowed" $1 million from Goldman Sachs. If he lost, how would he pay back the $1million? Why would Goldman Sachs finance his run? Ok, so Cruz won. Then he did his thing – trashing various other senators. I believed him at the time. I thought he was a constitutional powerhouse going after the corrupt establishment. What I didn't know | -1.919854 | -1.735408 | -1.829523 | -1.436872 | [
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was that it was all an act. Cruz and Goldman Sachs (the leader in the 2008 crash where $1.4 trillion was stolen from the public) conspired to do what Obama did and so far were successful. Now running for president – how did Cruz pay off his $1million loan? He used campaign contributions from his presidential run to pay off the $1million loan – is that ethical? Did Goldman Sachs give over $1million in presidential contributions to Cruz? If so, the $1million loan was an illegal bait and switch. So, now Cruz is running as an anti-establishment constitutional senator, but is he really anti-establishment or is he a liar and fraud? Being a stock market broker, I can tell you that the insiders make billions every time they crash the market, such as the market crash in 2008. When you have inside information, you buy put options and sell short to make money on the crash. This is why the establishment elite crash the market around every 10 years. Goldman Sachs has their former employees throughout the government – you can bet that is no coincidence. If you have any doubt that Cruz is owned by the establishment elite globalists, then watch the following video – they did all the research and present the facts.
You must watch this video, search youtube for V9K9Xedk5eM and you will see the video "Rick Wells vs Ted and Heidi Cruz and the one world government"
Sounds to me, he knows what he talking about, so you better pay attation. VOTE FOR TRUMP NOV.
3 2016.
Arlene Cole says
Trump is a white obama. All the time Cruz was filibustering in the Senate, calling McConnell a liar, and going over to the House to convince them to stop obama bills, Trump was donating thousands and thousands of dollars to the gang of eight, the clintons, reid, and pelosi. Where were you then? Were you paying attention to what was going on in our government and in our country? Or are you just an election critic who pays attention for a few months every four years?
I have been following Cruz for at least six years now and have investigated his background all the way to high school. The man has been a conservative his entire life. He is a moral man, and believes in our Constitution and understands how and why it was created and what it means. He is a man of integrity and honest character. Go to ConservativeReview.com and check out his votes and his standings. Senator Cruz is the ONLY true Conservative running, and the ONLY one qualified to be President and Commander-In-Chief!
Kathy Walker says
Give me a brake Cruz (Rafael Cruz) is a Senator a lawyer. He has tried to pull this off that he is not establishment for a long time . He is part of the Bushes and so is his wife . That's where the met He had duel citizenship with Canada where he was born until 2014. I voted for Trump in Florida and everyone else I know voted for Trump. Cruz needs to go back to Canada .
Cruz is not establishment? Then why is a Bush backing him?
The writing is on the wall……more lies soon to follow.
Agnes Tillerson says
The establishment despises Cruz because he is his own man and does not kow tow to their political ploys. If jeb Bush is backing him, its because Trump and Rubeo were running their mouths about brother George. I consider that fair play. Cruz is the only candidate that will return our nation to its origin that made it so great and envious by the world. He is a man of honor and integrity. Unlike Trump, he is a constitutional conservative who will not change policies and laws by whim. Unlike Trump, he is also a gentleman whose personage alludes to respect.
Trump Supporter says
Sorry to burst your Cruz bubble. Cruz is in the establishment he even met his wife when they worked on the Bush campaign together. There is a history there. Just because he vptes no to everything does not make him an outsider. The Republicanparty does not want TRUMP in office because the gravy train will end and he will uncover all
the cocorrupt ion. The GOP knows Cruz will lose to Hilary and itit's back to politics as usual. ….
to Trump Supporter:
Consistently voting "no" makes Cruz an outs | -1.232277 | -0.84457 | -1.555609 | -0.618741 | [
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ider in the most important way…and hobnobbing socially with Washington insiders doesn't necessarily invalidate his actions on the job
Killerjoe says
Not true, Barbara. The most support Trump has been able secure has been 40-45%, that leaves 55-60% of the people who want someone else. Trump's support is remaining steady at best, while Cruz's support is on the rise. If Rubio and Kasich, both establishment candidates, leave the race where do you think their support will go?
Killerjoe they will split again as a lot of them will know Cruz for what he is and that is a life time politician groom by the Bush dynasty and funded by the same Texas money makers. Rubio people are going to remember Ted Cruz Hawaii email scam saying Rubio is pulling out of the race and really same crap Cruz pulled on Ben Carson. Cruz is a tool that will be used by the same super packs who are funding him. All those hit adds they putting out about Trump is going to cost Ted Cruz in the long run.
Trump2016 says
Killerjoe (pathetic screen name), By your childish logic, 75-80% of Republicans want someone other than Cruz. Rest assured that all the blue collar Democrats that have switched Party affiliation did so to vote for Trump. No one is flocking to the polls for this Cruz clown. The only reason that any of these establishment Republicans even consider supporting Cruz is because Trump will shake up the establishment and no longer will they have the influence they demand behind the scenes. Trump cannot be bought. Cruz has already shown he can be bought.
Lastly, do you really want a candidate who puts a Chinese company ahead of American workers. Cruz did just that when representing the Shandong Linglong Tyre company, The company stole the patent of American company, put American workers out ion the street, and "Conservative" Ted chose to support the Chinese criminals over American workers. He will never step foot in the White House.
I peg to disagree, yes I looked at Trump, I considered him seriously as a possible Presidential Canditate but after long consideration, thinking logical about the candidates left for a conservative to choose from, Cruz is the logical answer. Trump is not a conservative, he is a man who uses the system to milk it. He is for Trump, period and to hell with everything we as conservative stand for. Don't forget as a deal maker his only principle is to win at all cost. Listening to the accusations and insults he passes out it is like a street fight, not a Presidential election.
It is sad that so many take sound bites thrown out during debates to feed their anger forgeting their principals and integrity.
Following the "Prudent Man Rule" Trump is not acceptable as a potential Canditate.
You speak of principals and integrity. Cruz will roll over faster than a turd in a hurricane. He would be worse than Obama – ok, maybe not that bad, but he is 100% owned and it will be business as usual if he is elected.
What does the "Prudent Rule" have to say about Cruz representing a Chinese company that that stole blue prints from an American tire company and put American workers out of a job? I'm just curious, are those the Conservative Values you see in Ted Cruz?
Take a close look at the Shandong Linglong Tyre Company. No righteous American citizen should ever cast a vote for Ted Cruz unless you are happy to see the Chinese steal American jobs. Cruz is nothing but a phoney hack politician just like Jeb Bush.
Those Democrats who are voting for Trump are setting us up for a royal flogging. They know that Hitlary has their nomination so they're setting us up with a sorry candidate, Trump so they will have a cakewalk for her in November. It is so obvious but too many of you are helping those Democrats put another Democrat into our top position. Trump hyas never been anything but a Democrat. He has given millions to Hitlary, Reid, Pelosi. Schummer and others. Please check THE DONALD out by using the internet for something other than telling we who have, how great he is. HE IS NOT.
He gave to the republicans and the girl scouts also…..he gives to everyone………..they love to be bought and he is the one shining the light on it so don't act like he does all this by stealth……………there are no limits to the stupidity of people | -1.393899 | -1.653039 | -1.969016 | -1.046016 | [
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worldwide.
Your right their votes will go to Cruz. They want Cruz to beat TRUMP, then go up against. Hillary and loose. Remember this there is no democratic and Republican parties. It them the Washington establishment against us the American people. They are all in bed with the same llobbyists and special interests. Big money donners don't care who they pay off just as long the corrupt politicians do what they are told to do!
You hit the nail on the head . They are all crooks Dems and Repubs. Are all fighting Trump Because they want the gravy train to continue. Lawyer crooks fighting we the people. That is why they would sell there mothers to get into Congress they all come out millionaires. They want Immagration and bad trade deals Congress and business profits . They don't care about we pions. Voted Trump in Florida
Thomas E Stanley says
I used to like Jeb, wouldn't vote for him as President against Trump, but think he could follow. Well not anymore.
In my mind Cruz is a crook and a liar, actually I wouldn't believe him if he said good morning. It's true he's better than Hilary and Sanders, but not much better than Obama, or maybe he's worse than Obama. No longer in Jeb's corner, he can fade away for all I care. He's too gullible for office and too establishment.
Agreed. He is the reason toilet paper was invented
another problem cruz has going against him, as far as i'm concerned, is he's from texas! I also don't like people that have to cheat to win.
Good one! Actually the article spoke of the Bush Dynasty, but they were being nice – it is really the Bush crime family.
wallace says
Nazi bush wackers …..
Stephanie Wilson says
you bush haters are disgusting human beings!
With Rubio OUT there is Trump and Cruz left and Bush is now backing Cruz. Do you want 4 more years like we
had when Bash was in you better voe for Trump ad not for Hilleray the LIER AND THE BIG ONE!!!!!!!!!
Sharon Jenkins says
Of course the GOP establishment is against Trump? Why? He can't be bought and he doesn't take kickbacks like our present GOP. He's not a friend of big Pharma, Oil & Gas, Foreign interest, Lobbyists. He is his own man. I like Cruz, but I don't think he can beat Hillary. Trump is the only candidate that can beat Hillary and that is what the Rep need right now. If Trump wins Florida and Ohio and the GOP nomination, the GOP establishment need to rally around the candidate. If not, Trump will bolt and go Independent (not a bad idea) and will pull Dem and Rep votes and could still win the Presidency. You may not like Trump, but he beats better than what we have had for the last 8 years. Nothing but lies and deceit.
Bennye says
Ditto to all of he above!!!
Bert Cundle Sr. says
Check the polls. Trump can't beat Hillary. Cruz can.
Cruz is not even beating Trump, how could he possibly beat thunder thighs. Stop believing in a propaganda poll. And stop thinking Cruz will do anything for the people.
DISGUSTED says
You Hit the Nail On the Head!
The Bushes want Hillary in the White House!
A vote for Cruz is a Vote for Hillary!
President Trump 2016!
Trump & Oprah?
And I have some swamp land I can sell you. Really Trump is against 3 now on Repub side Trump will make mince meat out of Clinton she is just a stupid crook she should be under the jail and Bill should be there too he's a pervert.
TELL IT LIKE IT IS Sharon!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Well… Bush Dynasty on the side of Cruz, should help Rubio! They act like they are still running things… LOL
The Bush Dynasty really don't want Cruz in the White House, he has served the Establishment just fine in the Senate! The Bushes want HILLARY! SHE HAS BEEN PROMISED THE WHITE HOUSE AFTER OBAMA by the Establishment Globalist (Remember George | -1.815759 | -1.846427 | -2.253075 | -1.530521 | [
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] | 1,332,898 | 5,120 | CommonCrawl | 18 | 15 |
W. Bush and his New World Order Speech?)
They know Trump is the only one that can BEAT HILLARY and that is the reason why they are spending HUNDREDS OF MILLIONS to try to Stop Trump!
Robert Early says
This will certainly not help Cruz. I was prepared
to support Cruz if Trump faultered. No longer. It's now Trump or no one.
Theresa brex says
Its Cruz, or I stay home. Will never vote trump. . . Ever!
We don't need the vote of stupid people. Whatever your reason to hate Trump you should still consider what is best for our country. If you went to a doctor for a heart transplant and he was the best would you turn him away because you didn't like him? I don't like Cruz, but if he was not owned by the establishment then I would consider him, but he is worse than just owned.
Thank God, there are people out there thinking for themselves!!!!
Trump for President 2016!
I don't trust trump to do anything he says. He changes his positions as often as we chbange our underwear. He has been caught in so many lies ( press conference with sons wine, NY spring water that will put anyone's label on it and bush brothers,not trump, steak.
He is the donor class of the establishment.
Ted Cruz however doesn't change his mind sentence by paragraph. He is stable and consistent. He will back our military and untie third hands. Finally under Cruz, Obamas fake offensive against Isis will end!
Vote CRUZ
VOTE TRUMP!!!!!!!!!!!!!!!!!!
A VOTE FOR CRUZ IS A VOTE FOR HILLARY! JUST WHAT THE BUSHES WANT!
NO ONE should be surprised, Cruz has LONG been in the Pocket of the Establishment and is for ending America's Sovereignty just like the Bushes! Just like his wife, who is a member of the Council on Foreign Relations and voted to Merge America with Mexico and Canada! If you didn't want to Admit it, you don't have any choice now! Let's not forget George W. Bush and his NEW WORLD ODER SPEECH, for those who have not studied this, it means the End Of America!
Want to know the Truth, watch these!
https://www.youtube.com/watch?v=2Kstyj5BTcM
https://www.youtube.com/watch?v=VtlO39wIRWs
The Bushes know Ted Cannot Win Against Hillary, That is their MAIN GOAL, HILLARY FOR PRESIDENT SO SHE CAN FINISH WHAT OBAMA STARTED!
anita says
since Bush lost ,now he wants to be chummy with Cruz and be his vice president if Cruz wins ,that is his strategy now ,
S. A. Moose says
We the People, include quite a large number of Ted Cruz supporters… So let's get the field down to the two of them and see who really wins.
Cruz said TRUMP voters are low informed. That's political for saying because I'm voting TRUMP i'm stupid. Welll i'm ia lot smarter then he thinks, I see right through his lies. I wouldn't buy a used car from him……
From your comments you are pretty dumb
Rose says
If Ted Cruz was real MAN a winner he wouldn't need the BUSHE's help
It's all so transparent that they are funding CRUZ , the establishment. Has there own agenda. And it's not for the people.
Trump is a phony pig. The real Republicians will stay home rather than vote for this piece shit
That has broke his marriage vows 50 or 60 times. Brags about it. Hillary will be President because of dumb ass holes like you
Our only hope is Cruz
Lisa Miller says
I thank God for Senator Cruz! He has had to fight the establishment, fight for the constitution and fight for what he was elected to do. I believe he will do what he says or go out fighting. He has a track record to prove it. He has been tried and tested. I have more confidence in him than Trump. Trump is a wild card. Trump vacillates and changes his opinion so much that I am not sure what he believes in or even what he will inst | -1.963684 | -2.046977 | -2.288583 | -1.533122 | [
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] | 1,332,898 | 6,144 | CommonCrawl | 18 | 15 |
igate if elected. How can we trust him? Trump has no record and he keeps giving us cliches. We need to check our anger at the door and vote for someone who won't bow down and cower before the opposition. We need someone who will fight for us as our president without grandstanding and being someone who looks more like they should be in a tabloid than the Oval Office.
I am ashame you are a Republician !!
Norma says
Have you notice there only one person that want to make sure we get our
Constitution and the Bill of Rights back to the American people. Obama didn't
leave much left of our Constitution. Trump busy using all the loop holes to
make himself richer, he has admit it at the debates and yet most of you are
will to vote for him……………..WHY??????
The Bush family will support Cruz just long enough to put Trump down. Then, Cruz will get the old RINO knife in the back. They are not about to elect an upstart Conservative.
How hypocritical of the GOP. Listen to the people not your special interests. We the people in the majority want Trump and there are more, once they stop feeling guilty about betraying their chosen candidate and see through all the manipulation and lies will also vote for Trump.
spike says
UH ?? The fact is Trump has not achieved anything close to 51% of the GOP vote let alone the general electorate.
So don't be so loose with the "majority" term. We the People need a candidate that can pull in 51% plus in General Elections or we will face at least four years of Hillary / Obama policies and worse. I just saw her promising NO deportations of illegals and a path to citizenship for ALL of them. We the People better get our heads out of our collective asses and support Trump and Cruz. Whichever one wins the nomination will have 100% of my support.
In fact I'd still like to see a Trump/Cruz ticket.
A vote for the people is a vote for Trump and a vote for Cruz is a vote for the corrupt establishment that includes the Bush and Clinton crime families. Cruz should be cancelled from everything and no way a VP.
Deb. Del
A vote for Trump shows how dumb you are
I was with until You said Cruz /Trump. Trump would drag the ticket down to much baggage.
Cruz/Rubio. Cruz / Kasich We need experience that can work congress. Without congress you are dead in. The
Water. Congress hates Trumps ass. They will not work with him. He would get not done Probably be impeached for war crimes this is not a good man
Yes, the term "majority" is still a questionable one. So, I ran my own poll of 400 Republicans in my network of five states. 78%
favored Trump.
Not really scientific, huh?
Lois Wenk says
We see Trump for what he is a fraud.
You've been sucked in by the propaganda.
Deb dell
That is about the dumbest post on the page no one met with Golden Sachs. You are one dumb ass to spread crap like that
he is like true politician ,two faced ,self serving like all of them are
Lois thank you for 8 years of Obama, keep voting for Hillary, she tells the truth and only wants to work for the people. I have a bridge in NY that you may want to buy…..The Clinton machine loves sheep's like you.
Trump supporter.
I voted for Romney. Trump is lieing piece of crap He does not know what he is talking bout No. Experience period !!
How hypocritical of the GOP. Listen to the people not your special interests. We the people in the majority want Trump and there are more, once they stop feeling guilty about betraying their chosen candidate will also vote for Trump.
I surely hope so. I hope people will stop listening to the establishment and look at the FACTS. NO more establishment. Its killing America.
Richard says
Loser backing another Loser.
How can the REPUBLICAN PARTY WIN
Get united behind the TRUE leader, TRUMP!
or the Republican Party will never win.
cruz and rubio and the nobody from Ohio
MUST DROP OUT NOW!
Should drop out
Main Street says
Mean old Mr. Trump upset the Jeb | -1.492516 | -1.950283 | -1.829523 | -1.64238 | [
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coronation. So what that Trump is getting tens of millions of votes. Don't they know that the ruling class demanded a Hillary vs. Jeb November election? Then I saw on the news how they have a commercial on stating that Trump never served in the military. Neither did Marco, Ted or Jeb. The retired military officers in the commercials probably owe the Bush family for their promotions. Yes men with stars on their shoulders.
They were generals before the Bushes You dumb ass. Most come out of Vietman. That's the war Trump got 4. Deferments.
From that. Ran like the coward he is
Redfray says
The second best reason I know to NOT vote for Cruz.
Gee, the Bush's promoting an unqualified candidate who was a FREAKIN' CANADIAN less than 2 years ago.
i'm shocked, shocked i say, that the Bush Crime Syndicate family would promote Unconstitutionality in any way(NOT!).
The Bush and Clinton family need to go away. Two crime families. We need Trump to clean up the mess in this country. Can't wait till the Obamas are gone too. Sick of all these leaches.
The corrupt establishment elite parasites need to go, Trump is the can of Raid.
Yes he is poison To the country
Andrea Guarneri says
I also think the republican super pacs and Jeb Bush is a hypocrite. Marco, Ted and Jeb HAVE NOT SERVED in the military.
Also, If Trump is leading in so many places, then the republicans should STAND WITH TRUMP. I think that he is THE ONLY PERSON THAT COULD BEAT HILLARY CLINTON.
He is a MUCH BETTER CHOICE THAN HILLARY.
Furthermore, he is a businessman and I think it is time for a businessman and a successful businessman to be in the WHITE HOUSE as president.
I totally agree Trump needs to clean house. He will be the best person for the job. Hillary is a lawyer crook she thinks she can get away with anything. Can't stand her.
I AGREE! THE DEMOCRATS AND ALL THE POLITICIANS HAVE TO GET OUT! THE WAY CLINTON LIES AND ABOUT BENGHZI HOW COULD ANY BODY THINK ON EVEN CONSIDERING HER! SANDERS FREE,FREE,FREE,IF THESE KIDS THINK THEY ARE GOING TO GET FREE COLLEGE, THINK AGAIN! THEY DO NOT EVEN KNOW WHAT SOCIALISM IS! (communism) WAKE UP CHILDREN!!!!!!!!!!!!!!!!!! THERE IS ONLY ONE PERSON WHO CAN GET OUR COUNTRY BACK! AND THAT IS DONALD TRUMP!!!!!!!!!!!!!
Trump ran like hell Got 4 deferrments during Vietnam. A real coward
Norman shatto says
Just proves what I've always known that Ted Cruz is a total sellout to the ruling class and will do whatever they want no matter how bad it is for the country. We need Donald Trump for the job at hand
Paul Meierhenry says
Go Cruz
Cruz lies Evan about his name (Rafael Cruz). He had duel citizenship with Canada where he was born until 2014. What he did with the robo calls with Trump Carson and Rubio shows what a scumbag he is.
Cruz is still a legal citizen even though he had dual citizenship. He was born in Canada & has an American mother, as Obama was born in Kenya and has an American mother. That's not an issue with me. I know Cruz is for American Freedom and Liberty, whereas Obama is for Socialism. Socialism has never worked and only destroyed people.
Obama is also for ISIS or ISIL as he prefers to call them.
V. Carrera says
I understand that all of our republican party is mad. So am I. My kid lost his job along with 200 others due to Obamacare and were told that was the reason as they couldn't afford the insurance. My friends got overlooked for jobs because the jobs were given to 3 somalians at Home Depot, | -2.122566 | -1.832102 | -2.293655 | -1.504507 | [
-1.832195222539856,
-2.41293760443533
] | [
-1.6029013941210848,
-2.061302310936373
] | [
-2.100900755924817,
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-1.285992292965865,
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forget the americans. I read the Associated Press on the horrible death of Mr. Stevens including I believe Sodomy..It made me weep as I imagined him all alone in Benghazi. No Republican should vote for Hilary. I will vote for any Republican over Hilary or Sanders. I will vote for Trump or Cruz or whoever is the frontrunner against a Democrat!
I too am under employed at home depot do to Obama Care. But it looks great in Washington it gives them bragging rights to say the unemployment rate is under 5%. I have to work 2 jobs to pay bills. I don't get a government check and free healthcare like all the politicians do.. also I don't get a governmen t credit card to fix up my house like Marco did…….This whole system is rigged against the American public. …..
Yes go Cruz all the way back to Texas and then if we lucky back to Canada.
My husband is retired military we both have voted for Trump. All of the other military people we know around the States are voting Trump. We are tired of the rest of the same lawyer crooks that are running this country.. We need a business man to clean house. All these other lawyers should be under the jail. Vote Trump 2016!!!!!!!!!!!
Trump targeted vets in his Trump University scandal. All military personnel are potential vets. You husband was slandered by virtue that is his potential. Unless you believe he is 100% going to die before his service to America ends. Trump slanders all you are mistaken to vote for Trump. Your husband and other present and past military personnel deserve better.
Merril says
I agree with these comments. The Republicans are going to keep their crap going and Hillary will be president! Then I would change my party affiliation also! Don't be stupid Republicans!
Hugiaino says
Hillary will be President if Trump gets nominated.
I don't think so……..Hillary will be going to the big house, not the White House.
R Davis says
Just as the people rejected Jeb, any association to the Cruz ticket could taint it. I believe the Bushes want to serve our nation, but their hard-line stance on common core and other things makes many believe they are associated with the one-world government. That change is repugnant to many. I am a Cruz supporter but if my background would taint the Cruz ticket, I would not make it a public announcement.
The GOP is getting pretty BAD! Like I've allways said. They are afraid that if TRUMP goes all the way, Their pockets are going to SHRINK! OH WELL! WE DO NOT NEED POLITICIANS! DONALD TRUMP IS THE ONLY ONE WHO CAN TURN OUR COUNTRY AROUND! AS FOR LITTLE MARCO AND LYIN TED! WE DO NOT NEED FOREIGNERS IN THE WHITEHOUSE! AND FOR BIG MOUTH KAISCH HE SAYS A CONVENTION! THEY ARE ALL SICK!!!!!!!!!!!!! DONALD TRUMP FOR PRESIDENT!
No, they're afraid that if Trump gets nominated, Hillary wins! Why? Because many polls of likely voters say so!
The best thing Trump could do for America is look at the polls, realize he can't beat either Hillary or Bernie, and support someone who can, like Cruz. If he'd clean up his mouth, lose the insults, and talk substance, he could make a positive difference in America.
Trump will not choose it.
Sam Mc Connell says
You folks just lost to the democrats and actually destroyed the republican party – you remind me of a phrase "HouseMouse" that was a term used when I was in the service. That is those suck asses that think they control all of us simple people. They had a bunch of them down in Georgia -McConnell Romeny and lots of lobbeyest – where you get more favors.
Romney, Paul Ryan, MConnel, are old boy crooks that need to go. Rubio Cruz Kasich Are all part of the lawyer crooks on the gravy train. Hillary and Obama are more of the same all eval crooks. Then we have Sanders a communist. This is what we have running this country if we pions did the stealing that they have done | -1.837674 | -2.086371 | -2.3038 | -1.595555 | [
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we would be in jail. That is where they all belong. Trump a businessman 2016. Our last hope before we go over the cliff.
Kathy H says
Of course they are rallying around Cruz although they would prefer the favorite son Jeb. This just confirms for me that Cruz is not as he portrays himself.
If the GOP continues their tricks and stop Trump, and if he is the popular choice – It is time to replace the Republican Party with an Independent party.
Ralph L Carpenter says
Maybe the "Conservative" party? Conservatives do not have any representation in todays two-party system.
That's the problem. Us real conservatives were left behind if Trump gets nomination. Nice having it explained.
Lorraine E says
Certainly Ted Cruz knows the history of the Bush's oil mafia connections. The only reason I can imagine Ted Cruz would do something so stupid is because the Bush's have unlimited money from their oil mafia connections. The Bushes all have very tarnished backgrounds and Ted Cruz's ties to them is enough reason to not consider Ted for the presidency.
This is another reason why only Donald Trump should become our next president. The Cruz/bush connection in my opinion disqualifies Ted Cruz for the presidency. We don't need another Bush puppet.
Lorraine-
And don't forget the Carlyle Group, either; that Corporation has fingers in a lot of pies.
If you haven't heard of it already, you may wish to read "Family of Secrets," by investigative journalist Russ Baker; if you do, the info you'll receive will soon disabuse you of any notion that the Bush family cares about America or Americans.
Oh, and i forgot; visit familyofsecrets.com, which will be much faster(and far less expensive).
I am so sick of all the crap going on to stop Trump. Just let the American people choose !!! I am ready to leave the Republican party right now. He is the only one that can clean up the mess in DC. If Trump gets the delegates and they cheat him out of the nomination, there will be rioting in DC. I am all for doing away with Republican party and replacing it with a new party. They don't do anything to help us anyway. They just roll over and let Obama have what he wants. How about the Patriot party ?? It will be for all of us who have been forgotten by the elites. We need to keep Mr. Trump in our prayers.
Trump should have gone on what he is instead of being a fraud. He is a fraud, he is not really a republican in any shape or form. Disrespectful to claim to be anything you are not. If he won on honesty like Sanders we would not be as mad at him. Trump is no better than H Clinton and Obama. We are not for voting for Sanders but he has not set out to be a fraud.
G-ma68 says
Establishment or not Cruz is still at the end of the day… A politician. Even if he is clean on the inside he seems slippery on the outside. Bushes' getting behind him will only solidify that slime. Too bad.
G-ma68:
At the end of the day, what he REALLY is is a CONadian who didn't give up his citizenship until 2014.
And for the poster that says he held "Dual Citizenship," check your facts; Canada did NOT ALLOW dual citizenship when Cruz was born.
President Ted Cruz would NOT get everything he asks Congress for. That shouldn't surprise anyone. President Obama governs by executive order because he is unable to get most of what he wants.
It's time for the non-Trump Republican candidates to meet and develop a strategy for dealing with Donald Trump. That would likely mean two of the Trump challengers getting out of the race and endorsing the remaining challenger. I think Gov. Kasich would probably be the best qualified of the three, but he has had problems winning primaries. At this point the object is to win the primaries.
gordon miller says
This is good news because a solid front is needed to defeat the witch and that front, the polls show, cannot be led by Donald Trump; he losses to her. A Cruz coalition is our only hope.
Gordon-
Polling corporations get hired, right?
"He who pays the piper calls the tune," right?
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] | 1,332,898 | 10,240 | CommonCrawl | 18 | 15 |
ing for these "polls," and you'll have a clue-by-four.
Face facts; Cruz was a Canadian in 2014, and is NOT QUALIFIED for the office of the Presidency. Period!
What worse, though, is that in 2012, during the run-up to the last D'ohbama election, Cruz personally pointed out that since D'OweBama didn't have 2 citizen parents, he wasn't natural born, and so was disqualified for the office – yet here he is, KNOWINGLY trying the same bullshit! He should be arrested for attempting to perpetrate a FRAUD.
The constitution says otherwise. He is a natural born citizen based on mother's citizenship just like Obama is. Read the constitution and get over yourself all birthers. This was already decided. Go for word.
Lois-
To put it mildly, your opinion is horse-hockey.
The Constitution does NOT SAY what "natural born" means. In the First Congress, Second Session, the LAWMAKERS defined the word the same as Vattel did, in his book "Law of Nations," which is that "natural born" status is ONLY conveyed if BOTH parents were Citizens at the time of birth, and that this principle applies to ALL nations, not just America. IF you disagree – since you keep saying that "the Constitution says," i say prove it. Google you up a copy, read it, then SHOW ME. Cut'n'paste the EXACT phraseology that you CLAIM is in there. I'll save you some time. It AIN'T IN THERE.
In addition, there have been at least 4 SCOTUS decisions involving the definition of "natural born," and in at least one of them Vattel is cited.
Ignorance can be cured; defiant stupidity merely awaits an opportunity for a slap-down. Educate yourself, woman.
I would think an endorsement from the Bush family would be the kiss of death for this election cycle. If Cruz desires to keep his "creds" he should reject their endorsement.
Charlie says
Follow the money!!!!!!
The lobbyist spread money around to career politicians, there is no end to it.
Donald Trump is the only candidate who is ready willing and able to take on the task of being president when we all know the whole world is in trouble like we have never seen before. He puts his money where his mouth is.
ALSO REMEMBER THIS: Everyone's life matters. It is how you participate in life that counts.
VOTE for somebody or shut the hell up
Bill capanna says
If the power people with all the money turn there backs on the people who voted Donald Trump in to be Our president then there should be a huge invest in these crooks . How they control our country and fill there pockets with all that money and give back nothing to the people who really need it is a crime .And then Cruz the Canadian does nothing but lie and be the Sleeze that he is .Rubio is no better and the Governor talks about what he has done and the country is still messed up .I will support Mr Trump because he is not afraid to confront the crooks like the republican senator who rules the senate by saying Trump will never be president I will make sure of that. Who in hell does he think he is this Kentucky senator could not stop Trump in his own state . Mr trump keep doing what you are doing we will support you against these Crooks
Getting endorsed by the Bushes is like being endorsed by Hitler or Charlie Manson.
Run away from the Bush clan as fast as possible, Ted Cruz!! Their support will be the death of your campaign. Compete honestly, and if the voters choose Donald Trump, then maybe he'll consider you for an upcoming Supreme Court vacancy. And this could be Mr. Cruz's best contribution to our nation.
No. If Donald wins the nomination, Ted needs to stay in the Senate to vote against all of Hillary's radical agenda!
Paul Bouffard says
I AM SICK & TIRED OF SEEING THE ESTABLISHMENT HIT ON DONALD TRUMP !THEY ARE AFRAID THAT IF HE IS ELECTED THEY WONT BE ABLE TO CONTINUE TO MAKE THEIR BACK ROOM DEALS $$$$$$ TO THEIR ADVANTA | -1.336373 | -1.246566 | -1.755972 | -0.964479 | [
-1.3007611560883368,
-1.3719842783962723
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-1.6960140803491903
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-1.5530726944418152,
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GE ? THATS WHAT THEY FEAR HIS BECOMING ELECTED ! I WAS A DEMOCRAT MOSTOF MY LIFE BECAME A REPUBLICAN WHEN OBAMA WON THE PRIMARY OVER CLINTON , , IF THE ESTABLISHMENT TRIES TO PUSH DONALD OUT NOW OR @ THE CONVENTION MYSELF & 100s OF THOUSANDS WILL VOTE FOR THE DEMOCRATIC NOMINEE ! WE WILL HAVE 4 MORE YEARS OF THE SAME BUT WE WILL SHOW THE BIG WIGS WHO HAVE OR ARE MAKING A LIVING IN POLITICS! A LESSON THAT THEY WORK FOR US &WE DONT ANSWER TO THEM ! ! ! I HOPE WE CAN SEE OUR WAY TO TERM LIMITS & NO MORE ( F U L L R E T I R E M E N T ) AFTER 1 TERM IN OFFICE ! DOES ANY OF YOUR EMPLOYERS OFFER YOU FULL RETIREMENT FULL PAY WAKE UP PEOPLE ! ! ! !
gizmo says
Ronald Reagan would bite his buckles now for selecting bush, he was not a politician at the time, however we found out his whole family is part of the new world order, and now they are indorsing cruse, it tells me what cruse is.
Stephan Brown says
Why is it we never hear about Niel Bush, the ex CEO of Silverado Savings and Loan in Colorado in the 80s? When Kenneth Star got to investigating his loan portfolio, and the fact that $60,000,000 was missing, VP daddy and Reagan fired Star and Neil disappeared. The money didn't, in was used to put brother GW in the White House and Jeb in as FLA governor. I am a Republican and I vote for the best of the worst. Still better that the Democrats. Obama, need I say more?
I guess I could say Hillary, but we need a president who can concentrate on the job and not spend time fighting indictments and overseeing politically expedient assassinations. (The Clinton associated body count now stands at 47).
What is the missing son, he who shall never be spoken of, up to???
So I vote for Trump, both the good and the bad about him, its all public record, love him or hate him; respect him for his transparency.
"Now, rumors indicate that the Bush political dynasty may be set to weigh in."
My heaven that does sound like a formidable clarion call designed to have the Trump campaign trembling in their shoes, in truth however it is ludicrous attempt to re-energize a spent force since the Bush brothers already combined their efforts on behalf of Jeb, and he has capitulated before the onward march of Trump – to then talk of marshalling the other dead beat Republican candidates, already fallen or about to fall like nine pins before the trumpeting Trump brigade, just further adds to the futility of the pathetic Republican party fight back. They still have not grasped that the peoples' support for the maverick Trump stems in large part precisely because he is not a part of the established Republican party, and for the these party establishment figures to be seen to gang-up against him will simply add to his public support.
Bobby says
Here establishment goes again treating us like cattle no matter who we want for the presidency they're saying no you can't vote for Donald trump you're going to vote for The establishment you're going to vote for who we want you to vote for I can tell you this they better watch what they wish for
I will stay home or vote for queen Hillary or Bernie Sanders
I am sick and tired of the media, pollsters and the same old tired guests who appear on CNN, Fox and other major networks.. I could care less what they have to say about all this political crap. I have no respect for the ones who continually denounce Donald Trump and I certainly do not value their opinions. Just shut the hell up and let the people of this nation make their own decisions. I believe Trump will win the Republican nomination and become the next President of the United States. God Bless America.
The only answer I see is: VOTE TRUMP RE | -1.506212 | -1.688852 | -2.258148 | -1.265181 | [
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-0.8405502119447533
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-1.9108895101063568,
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GARDLESS!
Bush has just (thank God) politically assassinated Cruz. RUBIO is a tantrum throwing, lying crybabe and Kasich is scarey quiet … Though my gut says he might be a nice guy… Which is not what we need right now.
Clearly they do not get the message sad.
Brainwashed people love the uphorea of not smart. It's like being stoned to a drug abuser.
jimmy says
Shelton nails it below…..
audrea says
The Democrats are running scared. The Bush's going for Cruz! How desperate can you be?They are all so afraid of Trump because he doesn't have to grease their hands for any of their contributions. He's his own man. He's out for the betterment of this country, not like all the other politicians that kiss ass for contributions and promise pay back. Trump is a man of his word and will finally put this country on the right track after Obama has nearly demolished it. AND…we will have a drop dead beautiful, classy first lady!!!
We need functional not tyrannical president. Trump is the problem. If she is or is not classy, it still does not out weigh the problem still is Donald Trump as president is disgusting at best.
Donald Trump – Love him or Leave him. I am a Trumpian all the way. He's a successful businessman, surrounds himself with the best money can buy and is able to delegate responsibilities to his staff. I believe he has much to offer this country and will make a difference.
You are spot on!. Some one who is not owned by the establishment . Someone unafraid to say it like he feels it.
That's Mr Trump.
DustyFae says
Personally, l do not care who you believe or not.. Or who you vote for.. God is in Control
Dusty-
Once upon a time, a farmer had a section of his land near a creek that was all overgrown with weeds, trash trees, thorns, vines and all sorts of non-useful things. One day, he decided to clean it all up and plant it, as it was good bottom-land. So he cut down the briers and the vines and the trashtrees, hauled out the stumps, burned the debris, plowed and planted the area, and lo and behold! It produced very, very well. He mentioned how the area had done so well to his pastor of a Sunday, and his pastor promised to come out an take a look. The following Wednesday, the clergyman showed up. "Wow!" exclaimed the pastor, "i remember how this once looked! The Lord sure has blessed you!" The farmer began to fidget. The pastor walked through another area, and said "Ah declare, these are the largest tomatoes i've ever seen! The Lord has truly been good to you." The farmer began to fret. The pastor took an ear of corn off the tallest cornstalk he'd ever seen, and said, "The Lord has truly done right by you!" The farmer couldn't stand it no more, so he said "Pastor, the Lord had that patch all to Himself for many, many years. It wasn't 'til i did something that it became what you see today!"
Told you that to tell you this: there's a great difference between saying "The Lord is in Control" and "The Lord directed me to do A, B, or C, and look how it prospered when i worked at it." I have seen far too many use the phrase "The Lord is in Control" as an excuse to do nothing and leave it all in His hands. I certainly hope you lean toward the former and not the latter philosophy, for as the theoretical farmer in the example above demonstrates, nature, when left entirely to itself, does not tend toward abundantly producing but tends towards unprofitability.
Spartacus says
It's hard to believe Cruz is STUPID enough to want to connect himself to Bush….he's got to realize that the reason Trump is so popular is the national hatred of anything relating to the Bush/Clinton political machine…and of course anything that reeks of O'Vomit! I think Bush is trying to buy his way onto someone's ticket as a VP candidate….or maybe the family is so worried about Trumps promise to reveal the truths of 9-11 that they're willing to put $$ behind Cruz to stop | -1.035044 | -1.62797 | -1.720465 | -1.539626 | [
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Trump (and I'm sure Cruz isn't dumb enough to turn down free money for his Campaign). Trump might be great….IF he does the things he's talking about NOW and not the things he said he was for 10 years ago (we know we can't trust Rubio….man who speaks with forked tongue about Amnesty for illegals and 2d Amendment rights…Trump is someone we just can't be sure about). Cruz at least has been consistent and has fought against the RINOs and GOP "establishment" all along. I'll admit I'm intrigued about Trump….but my 1st vote will be for the guy I KNOW is on our side….Cruz (unless he picks Jeb to be his VP, then I might have to vote for The Donald!)
Trump is the only one that can get our country out of debt. He is a business man and knows how to do it. He's for the second amendment, veterans, security to stop people that are illegal from coming here. He is for every one and also said he will not raise the age limit on social security or touch it . He said he will leave social security the same. And when he says we are going to make America great again, he means it. We will have more jobs and you can believe that!!!! "Donald Trump for president 2016"
TammyP-
The curious thing is that in America – since at least 1964, the last time there was real silver in any Coin of the United States in general circulation – all money is debt-notes, and if all the debt notes were repaid(a mathematical impossibility, btw), all of what currently passes for "money" would disappear. But don't believe me; believe these guys:
"The whole profit of the issuance of money has provided the capital of the great banking business as it exists today. Starting with nothing whatever of their own, they have got the whole world into their debt irredeemably, by a trick. This 'money' comes into existence every time the banks 'lend' and disappears every time the debt is repaid to them. So that if industry tries to repay, the money of the nation disappears. This is what makes prosperity so 'dangerous' as it destroys money just when it is most needed and precipitates a slump. There is nothing left now for us but to get ever deeper and deeper into debt to the banking system in order to provide the increasing amounts of money the nation requires for its expansion and growth. An honest money system is the only alternative."
— Frederick Soddy(1877-1956) British author, professor, Nobel Prize for Chemistry, 1921
"Commercial banks create checkbook money whenever they grant a loan, simply by adding new deposit dollars in accounts on their books in exchange for a borrower's IOU."
— Federal Reserve Bank of New York
Source: I Bet You Thought, p.19
"If two parties, instead of being a bank and an individual, were an individual and an individual, they could not inflate the circulation medium by a loan transaction, for the simple reason that the lender could not lend what he didn't have, as banks can do…only commercial banks and trust companies can lend money that they manufacture by lending it."
~Professor Irving Fisher, economist, in his book 100% Money, 1935.
"The banks — commercial banks and the Federal Reserve — create all the money of this nation and its people pay interest on every dollar of that newly created money. Which means that private banks exercise unconstitutionally, immorally, and ridiculously the power to tax the people. For every newly created dollar dilutes to some extent the value of every other dollar already in circulation."
— Jerry Voorhis[Horace Jeremiah Voorhis] (1901-1984) US Congressman (D-Ca)
"This is a staggering thought. We are completely dependent on the commercial Banks. Someone has to borrow every dollar we have in circulation, cash or credit. If the Banks create ample synthetic money we are prosperous; if not, we starve. We are absolutely without a permanent money system. When one gets a complete grasp of the picture, the tragic absurdity of our hopeless position is almost incredible, but there it is. It is the most important subject | -0.315965 | -0.245605 | -0.55633 | 0.080377 | [
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intelligent persons can investigate and reflect upon. It is so important that our present civilization may collapse unless it becomes widely understood and the defects remedied very soon."
― Robert H. Hemphill (Credit Manager of Federal Reserve Bank, Atlanta, Ga.)
"When the Federal Reserve Act was passed, the people of these United States did not perceive that a world banking system was being set up here. A super-state controlled by international bankers and international industrialists acting together to enslave the world for their own pleasure. Every effort has been made by the Fed (Reserve) to conceal its powers but the truth is — THE FED HAS USURPED THE GOVERNMENT!"
― Congressman Louis McFadden, Chairman of House Comm. on Banking and Currency from 1920 to 1931 (Shadows of Power, pg. 23
"I am afraid that the ordinary citizen will not like to be told that the banks can and do "CREATE" money, and they who control the credit of a nation direct the policy of the government and hold in the hollow of their hands, the destiny of the people."
― Reginald McKenna, Chancellor of England's Exchequer, January 1924
"The present Federal Reserve System is a flagrant case of the Government's conferring a special privilege upon bankers. The Government hands to the banks its credit, at virtually no cost to the banks, to be loaned out by the bankers for their own private profit. Still worse, however, is the fact that it gives the bankers practically complete control of the amount of money that shall be in circulation. Not one dollar of these Federal Reserve notes gets into circulation and without someone paying interest to some bank to keep it circulation. Our present money system is a debt of money system. Before a dollar can circulate, a debt must be created. Such a system assumes that you can borrow yourself out of debt."
― Willis A. Overholser, LL.B – in History of Money in the United States.
These statements were made during hearings of the House Committee on Banking and Currency, September 30, 1941. Members of the Federal Reserve Board call themselves "Governors." Governor Marriner Eccles was Chairman of the Federal Reserve Board at the time of these hearings:
Congressman Patman: "How did you get the money to buy those two billion dollars worth of Government securities in 1933?"
Governor Eccles: "Out of the right to issue credit money."
Patman: "And there is nothing behind it, is there, except our Government's credit?"
Eccles: "That is what our money system is. If there were no debts in our money system, there wouldn't be any money."
Congressman Fletcher: "Chairman Eccles, when do you think there is a possibility of returning to a free and open market, instead of this pegged and artificially controlled financial market we now have?"
Governor Eccles: "Never, not in your lifetime or mine."
— Marriner Stoddard Eccles(1890-1977) US banker, economist, and Chairman of the Federal Reserve (1934-48)
Source: during hearings of the House Committee on Banking and Currency, September 30, 1941.
++++++++++++++++++++++++++++++++++++++++++++++++++
The short story is this: nowadays, Debt IS money – well, at least until the "loans" are called in. This is the major tactic of the "Beast that deceived the whole world." With debt-money, you can't really "buy" anything, because you're trading IOU's for goods or services. The only way to permanently extinguish debt is by trading substance for substance, that is, something of intrinsic value for something else of intrinsic value. With debt-notes(aka FRNs or Federal Reserve Notes, or dollar "bills," that is, BILLS for dollars, promises to pay), all you can do is "legally discharge debt."
If you want further information on this world-wide fraud, you might wish to start by googling up a court case called "Stanek v White," or you could get a clue by four from many vids available @ youtube, | -0.0541 | 0.857422 | -0.051618 | 0.878348 | [
0.06754459748142487,
-0.17574400059447992
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] | [
0.03207794642335302,
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1.003213543096673,
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Health & Medical Management
Paralegal & Legal Assistant
Liberal Arts & Humanities
Geological & Earth Sciences
Trade/Technical
Cosmetology & Related Services
Vehicle Maintenance & Repair
Featured Online Colleges & Universities
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Are You Serious About Going Back to School?
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The Fundamentals of Financial Aid
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Calculating the Cost of School
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Type of School Private, for-profit
4-year or Above
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50%) N/A
SAT Critical Math (Middle 50%) N/A
SAT Critical Writing (Middle 50%) N/A
ACT Composite (Middle 50%) N/A
See More About Admissions »
Bachelor of Science in Information Technology - Cyber Security (Online)
Grand Canyon University's Bachelor of Science in Information Technology with an Emphasis in Cybersecurity prepares you to enter the fast-growing, in-demand field of information technology. Gain an in-depth understanding of the value of IT, including how it can improve business performance, boost organizational efficiency and reduce costs of enterprise systems. The cybersecurity degree program focuses on core competencies about IT-driven business, software technology and tools, enterprise information systems and cybersecurity. The BS in information technology with an emphasis in cybersecurity exposes you to the relationships among cyber defense, cyber operations and cyber law. Acquire working knowledge of processes and goals of cyber forensics, and develop plans and strategies for security architectures. The emphasis culminates with an IT capstone project in cybersecurity. IT cybersecurity emphasis domains include cyber operations planning and execution, cybercrime and cyber law, as well as digital forensic investigations, security architecture, information assurance and secure system administration. This information technology degree program trains you to problem solve using technology and computer programming, while also leveraging your own experience. A focus on applied, hands-on activities ensures you acquire the proper training to excel upon immediately entering the workforce in cybersecurity.
Request More Information »
Bachelor of Arts in English for Secondary Education (Online)
Grand Canyon University's Bachelor of Arts in English for Secondary Education is an accredited degree program that prepares aspiring English teachers to meet the challenges of the modern secondary education classroom. Graduates who successfully complete state certification requirements will be qualified to teach students in grades 5-12. This degree is a joint program offered by GCU's College of Humanities and Social Sciences and the College of Education.
Bachelor of Arts in History for Secondary Education - ITL (Online)
The Bachelor of Arts in History for Secondary Education degree program at Grand Canyon University recognizes that middle and high school history teachers do more than teach facts, events and dates—they impart ethical standards and wisdom developed over centuries of human history. This history teacher degree, offered by the College of Humanities and Social Sciences, is regionally accredited and approved by the Arizona State Board of Education. Students take coursework that aligns with the requirements established by the National Council for Social Studies (NCSS).
Bachelor of Science in Accounting (Online)
Grand Canyon University's Bachelor of Science in Accounting program addresses the areas of servant leadership,innovation,entrepreneurial spirit,and business skills in order to develop the global citizens,critical thinkers,effective communicators,and responsible leaders required in today's global economy. The program prepares students for professional practice in corporate accounting,corporate finance,or public accounting. Completing the requirements for the accounting major qualifies students to sit for the Certified Management Accountant (CMA) exam and the Certified Public Accountant (CPA) exam.
Bachelor of Science in Applied Business Analytics (Online)
The Bachelor of Science in Applied Business Analytics is oriented toward students who have a business mindset and propensity for data and analytics. You will acquire data skills that relate to working with datasets to find meaning, along with analytics skills that comprise of defining and implementing models to gain insight into predicting what might happen or recommending a course of action(s). Through this transfer-friendly business analytics degree program, you will examine the fundamentals of database structures, data mining, analytics modeling. You will also observe the role of governance and ethics within information technology with specific emphasis on preparing for an audit, complying with government regulations and understanding data-privacy issues. Since analytics and technology are often intertwined, information technology project management is also addressed in the program. This BS in Applied Business Analytics degree program is specifically targeted toward the emerging business analytics profession for students who have an existing foundation in selected areas of business, such as accounting, finance, marketing and management. Emphasis is placed on a combined mix of database and analytical skills. You will learn to leverage core business knowledge and analytic skills to help gain insight | 0.275393 | -0.171294 | 1.799838 | 0.058916 | [
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into and/or solve business problems using an analytics approach.
Bachelor of Science in Applied Business Information Systems (Online)
Grand Canyon University's Bachelor of Science in Applied Information Systems degree program will examine the fundamentals of business information systems and technology, including programming, principles of database management, networking, IT project management and ethical considerations related to IT governance. The BS in Applied Business Information Systems program is designed for individuals who have a business mindset and propensity toward information systems technology. You will gain database, programming and system administration skills, as well as skills related to leveraging information systems to help organizations achieve success. This applied business information systems degree is designed to build upon an existing foundation of selected areas of business, e.g., accounting, finance, marketing and management. Skills developed in this program include computer technical, problem solving, strong work ethic and critical thinking skills.
Bachelor of Science in Applied Entrepreneurship (Online)
Explore innovation, cultivate an entrepreneurial spirit and hone your business skills in this Bachelor of Science in Applied Entrepreneurship degree program. Offered by the Colangelo College of Business at Grand Canyon University, this program examines the principles of entrepreneurship and builds a solid foundation for analytical thinking, problem solving and contributing to today's global economy.
Bachelor of Science in Applied Human Resources Management (Online)
GCU's BS in Applied Human Resources Management can equip you with the essential skills and knowledge necessary for a successful career in human resources. The degree program emphasizes how industries and organizations use human resources management to enhance their overall performance and success. Courses and assignments allow students to develop the ability to apply these skills in a variety of different contexts and across numerous fields and types of organizations.
Bachelor of Science in Applied Management (Online)
Grand Canyon University's Bachelor of Science in Applied Management program emphasizes organizational behavior, business management, servant leadership, marketing concepts, managerial accounting and effective oral and written business communications required of business managers.
Bachelor of Science in Applied Technology (Online)
Grand Canyon University's Bachelor of Science in Applied Technology degree focuses on problem-solving using technology with hands-on activities designed to provide students technical experience required in the workplace. Throughout this program, students take courses in networking, security, database systems and information technology process management.
Bachelor of Science in Behavioral Health Science (Online)
The Bachelor of Science in Behavioral Health Science degree program prepares students to work as behavioral health technicians or specialists in various settings as members of a clinical team. This behavioral health program within the College of Humanities and Social Sciences provides a more general and broader perspective on specific areas in the behavioral health field.
Bachelor of Science in Behavioral Health Science - Childhood and Adolescence Disorders (Online)
The Bachelor of Science in Behavioral Health Science with an Emphasis in Childhood and Adolescence Disorders degree program prepares students with the skills and training to work in various settings as behavioral health technicians or specialists. You will gain a comprehensive knowledge base of childhood and adolescent development and behavior, and in particular, a thorough understanding childhood and adolescent disorders.
Bachelor of Science in Behavioral Health Science - Family Dynamics (Online)
The Bachelor of Science in Behavioral Health Science with an Emphasis in Family Dynamics degree program develops graduates who are prepared to work as behavioral health technicians or specialists. These professionals may work in various settings, such as residential, inpatient or outpatient clinics, as members of a clinical team. This behavioral health science program within the College of Humanities and Social Sciences is designed to provide an extensive study of behavioral health with a strong foundation in family dynamics.
Bachelor of Science in Behavioral Health Science - Infancy and Early Childhood Studies (Online)
Grand Canyon University's Bachelor of Science in Behavioral Health Science with an Emphasis in Infancy and Early Childhood Studies program provides students with an introductive study on the development and behavior of infants and young children. Students gain the knowledge and skills necessary to work with young children and advocate the importance of the family structure and other caregiving systems.
Bachelor of Science in Behavioral Health Science - Substance Use Disorders (Online)
Grand Canyon University's Bachelor of Science in | -0.055726 | -0.320084 | 1.96723 | 0.259222 | [
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Behavioral Health Science with an Emphasis in Substance Use Disorders program introduces students to behavioral health and substance abuse disorders and addiction. The substance use disorder curriculum offers a strong foundation for students who want to pursue a master's degree in counseling, mental health, criminal justice, clinical or forensic psychology or human services.
Bachelor of Science in Behavioral Health Science - Trauma (Online)
The Bachelor of Science in Behavioral Health Science with an Emphasis in Trauma degree program provides students with the foundation to enter the field as behavioral health technicians or specialists. You will acquire the skills and training to work within diverse settings, such as an inpatient, outpatient or residential clinic, as part of a clinical team. This behavioral health science program within the College of Humanities and Social Sciences offers a comprehensive study of the biopsychosocial impact of trauma and trauma-informed care.
Bachelor of Science in Business Administration (Online)
Grand Canyon University's Bachelor of Science in Business Administration program addresses the areas of servant leadership,innovation,entrepreneurial spirit,and business skills in order to develop the global citizens,critical thinkers,effective communicators,and responsible leaders required in today's global economy. The program prepares students to compete for entry-level management positions in business through coursework that addresses the key functional areas of management,accounting,finance,marketing,operations,and human resources.
Bachelor of Science in Business Analytics (Online)
Grand Canyon University's Bachelor of Science in Business Analytics program, offered by the Colangelo College of Business, examines foundational aspects in business, including marketing, accounting, finance, economics and management. You will engage in the critical competencies of the university's mission, including effective communication, critical thinking, Christian worldview, leadership and global awareness, perspectives and ethics. Additionally, the bachelors in business analytics program teaches the core competencies and domains of business analytics.
Bachelor of Science in Business Information Systems (Online)
Grand Canyon University's Bachelor of Science in Business Information Systems combines a solid business foundation with the technical skills necessary to compete for entry-level management positions in information systems and technology management. Business coursework will address functional area skills such as accounting,economics,statistics,finance,and marketing. Students will also examine the fundamentals of business information systems and technology,including programming,principles of database management,networking,IT project management,and ethical considerations related to IT governance. Innovation and entrepreneurship will be a key focus of the program.
Bachelor of Science in Counseling - Addiction, Chemical Dependency and Substance Abuse (Online)
The Bachelor of Science in Counseling with an Emphasis in Addiction,Chemical Dependency,and Substance Abuse is designed to train practitioners in the field of addiction counseling,chemical dependency,and substance abuse treatment. The coursework focuses on the nature of psychopathological conditions and their impact on the physical,emotional,social,intellectual,and spiritual nature of the human being. Core courses include study of the psychopathological behavior's impact on the individual,family,and society; screening and assessment methods at the beginning of treatment; and case management during the treatment phases. Counseling skills courses and a fieldwork/practicum course complete the focused coursework list. Completion of this program may qualify graduates to meet the standards for state,national,and international certification for professional prevention specialists and treatment counselors/providers. Participants should contact the appropriate certifying body to ensure that courses meet certification requirements.
Bachelor of Science in Cybersecurity (Online)
Grand Canyon University's Bachelor of Science in Cybersecurity degree program was developed with industry guidance to produce highly skilled, well-equipped cybersecurity professionals. As an increasing number of cyberattacks and information security threats hit our nation, expert cybersecurity professionals are in significant demand by government agencies, defense firms, financial services and companies in a wide range of industries that handle private data. Students will gain knowledge in defending digital spaces, computer environments, networks and sensitive information from malicious software developers and hackers. This program teaches and assesses competency in all aspects of defensive and offensive | 0.008135 | -0.33245 | 1.90636 | 0.547974 | [
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cybersecurity, cyber law and cyber ethics. Areas of study include: information assurance foundations, digital forensic investigations, malware reverse engineering, wireless security, security architecture design, security frameworks and secure system administration. The program emphasizes critical thinking, real-world application and practical project management experience. Students further develop valuable workplace skills, such as effective communication, teamwork, initiative, strong work ethic, analytical skills, adaptability and self-confidence. GCU's Christian worldview is also integrated in the program, so students can learn professional and ethical practices associated with cybersecurity. This is a technical degree that incorporates real-world, hands-on application of theoretical concepts.
Bachelor of Science in Early Childhood and Early Childhood Special Education (Online)
Learn to recognize and help children overcome developmental delays with the Bachelor of Science in Early Childhood Education and Early Childhood Special Education degree. Channel your passion for educating young children while preparing for certification in early childhood education as well as early childhood special education in states where the latter is applicable. As the ability to recognize developmental delays early in life has expanded, so has the demand for special education-prepared teachers. Build your skills in classroom management and pedagogy while also learning strategies for family, community and clinical involvement in meeting the needs of exceptional children.
Bachelor of Science in Early Childhood Education (Online)
Bachelor of Science in Early Childhood Education (Leads to initial teacher licensure). An increasing emphasis on early childhood education programs,and a general movement toward universal preschool education,will increase the importance of and demand for teachers with expertise in this critical area of teaching and development. This relevant and robust degree program is designed for individuals seeking knowledge and skills in the field of early childhood development and teaching. The curriculum includes instructional strategies,teaching methodologies,assessment techniques,theories of early childhood growth and development,and the impact of family and cultural diversity on early childhood. The degree program has embedded practicum experiences throughout the coursework,requires a student teaching component,and,upon approval,will lead to a teaching credential in Arizona. Teacher candidates are required to complete their internship experiences in a K–3 classroom. Graduates of this program will be equipped with the tools,techniques,and knowledge needed to successfully motivate,develop and educate children from birth to Grade 3.
Bachelor of Science in Educational Studies - Does Not Lead to Teacher Licensure (Online)
The field of education plays a unique role in effecting positive change in communities and society as a whole. The Bachelor of Science in Educational Studies degree program offered by Grand Canyon University prepares graduates to work toward a brighter future by influencing educational policies, laws, programs and curricula. Students taking this educational studies degree program from the College of Education at GCU will critically examine and analyze pedagogical theories, teaching practices, education laws and learning environments.
Bachelor of Science in Elementary Education (Online)
Grand Canyon University's Bachelor of Science in Elementary Education degree program prepares students for a career as a K–8 elementary teacher and to seek initial licensure. Earning this degree serves as a pathway for entering the field of education to lead in children's mental, emotional and social development, as well as to provide students with a foundation in early basic academics (like math and reading). At this stage, teachers are responsible for identifying their students' strengths and weaknesses. Based on those insights, teachers will tailor content, activities and projects that meet diverse student needs and interests. Other job responsibilities include motivating students, establishing behavioral rules and communicating with parents. With a bachelor's degree, teachers will have the knowledge and tools to follow their passion for teaching and making a lasting impression on their students' future. The teaching profession in elementary classrooms is challenging; however, the rewards of seeing children understand something or make a discovery on their own makes all the hard work worth it. This program's curriculum further includes opportunities for relevant application of concepts, theories and research with an emphasis on elementary educational field experiences; teacher candidates complete 120 hours of observational and practice-based experiences. A 15-week student teaching experience further shapes students into career-ready individuals who can confidently enter the classroom. Graduates are eligible for a 120-credit elementary education teaching credential in the state of Arizona..
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Teaching jobs in all sectors, including Christian education, are on the rise. As parents feel more empowered to make choices in their children's schooling, they are finding schools that align with their belief systems and values. Many Christian schools are looking for people with Christian education degrees to meet those growing needs. This Bachelor of Science (BS) in Elementary Education with an Emphasis in Christian Education degree program at GCU will help you learn, not only how to be an elementary teacher, but how to do it from a faith-based perspective. You will develop the skills needed to guide children through an academic journey infused with a biblical perspective. In addition, you will study child and adolescent development in both theory and practice through class and field learning experiences.
Bachelor of Science in Elementary Education - English as a Second Language (Online)
Grand Canyon University's Bachelor of Science in Elementary Education with an Emphasis in English as a Second Language degree program prepares teacher candidates for teaching in a K–8 classroom, specifically geared toward working with students who are considered English Language Learners. Graduates may seek an elementary teaching license to teach these students in an elementary classroom setting. Throughout this program, students will examine adolescent development, instructional planning, educational trends, cultural differences/cross-cultural studies, student engagement and managing learning environments, while gaining a strong liberal arts foundation. The program concludes with the Student Teaching for Elementary Education with an ESL Emphasis course, which exposes teacher candidates to a full-time, 15-week student teaching component that includes practical ESL elementary classroom experiences. Candidates will engage in research, analysis and hands-on teaching to support the creation of a Teacher Work Sample.
Bachelor of Science in Elementary Education - STEM (Online)
STEM includes the areas of science, technology, engineering and mathematics that concentrate on understanding the world and solving problems to transform our everyday lives. STEM is the future of the global economy and the coalescence of various disciplines to innovate in the 21st century and keep our country competitive. Careers in STEM have been on the rise and continue to be in high demand as the world becomes more technologically advanced. To drive STEM, the fields are calling for more STEM-qualified teachers who can ignite interest, curiosity and excitement in STEM education, starting at the elementary level.
Bachelor of Science in Elementary Education - Teaching Reading (Online)
Grand Canyon University's Bachelor of Science in Elementary Education with an Emphasis in Teaching Reading degree program is designed for students seeking initial teacher licensure and who are preparing for a career as an elementary teacher. This program provides an advantage to elementary education students through additional reading courses to support reading diagnosis, remediation and intervention processes.
Bachelor of Science in Elementary Education / Special Education - Dual Major (Online)
The Bachelor of Science (BS) in Elementary Education and Special Education degree offered by Grand Canyon University is designed for future teachers interested in working with children with special needs in the elementary classroom setting. The dual degree program includes coursework that prepares future elementary teachers (K-8), as well as coursework that is required of special education teachers working with mild to moderate disabilities in the K-12 setting. Both pedagogical knowledge and special education-specific content make up the curriculum for the BS in Elementary Education and Special Education degree. Graduates of the program are prepared to work with students with mild to moderate special needs by implementing individualized educational plans to accommodate students' various learning, behavioral and social needs. They also gain a well-rounded liberal arts elementary education background.
Bachelor of Science in Entrepreneurial Studies (Online)
Grand Canyon University's Bachelor of Science in Entrepreneurial Studies program addresses the areas of servant leadership,innovation,entrepreneurial spirit,and business skills in order to develop the global citizens,critical thinkers,effective communicators,and responsible leaders required in today's global economy. The program is built on the principles of personal integrity,values,and innovation. It provides students with the personal and business skills to think analytically,ask the right questions,solve problems,and function as entrepreneurs in their own business or intrapreneurs in larger companies.
Bachelor of Science in Finance (Online)
A Bachelor of Science (BS) in Finance degree from GCU prepares graduates to work in the financial world | 0.526386 | -0.328645 | 2.180274 | 0.204593 | [
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. Finance degree majors learn about industries including financial services, wealth management, corporate finance, insurance and real estate. Students learn how to understand/analyze data and information related to money management. GCU graduates who major in finance can integrate their understanding of financial markets on a global scale with decision-making for everyone from individual clients to major corporations.
Bachelor of Science in Finance - Financial Planning (Online)
A financial planning degree is a four-year undergraduate degree program that helps students learn about all aspects of financial planning from evaluating risk and investments to taxes, diversification and ethics. According to the U.S. Bureau of Labor Statistics, a financial planning bachelor's degree is typically required for financial advisor roles.
Bachelor of Science in Finance and Economics (Online)
Grand Canyon University's Bachelor of Science in Finance and Economics program addresses the areas of servant leadership,innovation,entrepreneurial spirit,and business skills in order to develop the global citizens,critical thinkers,effective communicators,and responsible leaders required in today's global economy. The program prepares students to compete for entry-level positions in corporate finance and in banking,insurance,and investment companies.
Bachelor of Science in Health Care Administration (Online)
Bachelor of Science in Health Care Administration. This professional degree program equips students for entry-level supervisory roles in health care organizations. Ideal candidates for this program are students looking for career entry in health care administration and those looking to advance from clinical or technical roles to supervisory roles. The program emphasizes both the conceptual and analytical skills required to manage in contemporary health care organizations. The program also prepares students who wish to eventually seek their master's degree in order to obtain senior health care executive positions. The curriculum features investigative and experimental opportunities in project management,teamwork,and leadership. Students of this program will have the opportunity to gain an in-depth understanding of the health care organization,strategic management and decision-making,finance,human resources,marketing,information management,and quality improvement. In addition,the impact of various dynamics,including regulatory,legal,ethical and public policy,are explored. Graduates of this program are prepared for administrative positions in hospitals,long-term care,outpatient facilities,physician offices,mental health organizations,insurance companies,public health agencies,and other types of health organizations.
Bachelor of Science in Health Information Management (Online)
A bachelors in health information management degree provides the knowledge, resources and tools needed to advance professional standards in health care and support delivery across many settings. GCU's health information management degree program prepares students understand the importance of information accuracy, information confidentiality and up to date analytics in providing first-rate health care.
Bachelor of Science in Health Sciences (Online)
This Bachelor of Science (BS) in Health Science degree from Grand Canyon University is designed for working professionals in healthcare fields who are seeking to advance in their clinical discipline. Healthcare professionals including respiratory therapists, radiology technicians, phlebotomists, surgical personnel, physical therapy assistants and EMTs, who have associate degrees or certificates in allied health can enhance their knowledge with this health science degree.
Bachelor of Science in Homeland Security and Emergency Management (Online)
The Bachelor of Science in Homeland Security and Emergency Management degree program highlights the application and understanding of preparation, leadership and communication in crisis contexts. The courses are structured to cultivate critical thinkers and ethical decision makers, prepared for disasters and emergency situations.
Bachelor of Science in Hospitality Management (Online)
Grand Canyon University's Bachelor of Science in Hospitality Management prepares students to compete for entry-level positions in the hospitality industry. Core business courses incorporate servant leadership,innovation,and entrepreneurial spirit,and address the key functional areas of management,accounting,finance,marketing,and operations. The major courses offer a broad-based curriculum specific to hospitality management,including hotel and restaurant management,event planning,facilities management,and human resources in the hospitality industry.
Bachelor of Science in Information Technology (Online)
Grand Canyon University's Bachelor of Science in Information Technology prepares students to enter careers in the rapidly expanding field of information technology. This IT degree program provides an understanding of the value of information technology and how this professional special | 0.129694 | 0.033285 | 1.977375 | 0.319053 | [
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ization can improve the performance, increase the efficiency and lower the cost of enterprise systems. Information Technology degree program domains include platform technologies, networking, information assurance, security and management, programming fundamentals, cloud computing, data centers, and Web systems and technologies.
Bachelor of Science in Math for Secondary Education (Online)
Grand Canyon University offers the Bachelor of Science in Mathematics for Secondary Education degree program to prepare aspiring math teachers to pursue a career inspiring and educating adolescent learners. Education is a field that is subject to continual change over time. With a modern curriculum and diverse learning experiences, GCU seeks to graduate teacher candidates who are fully prepared to meet the challenges of tomorrow's classroom.
Bachelor of Science in Nursing - RN to BSN (Online)
If you are a registered nurse with an associate degree or diploma certificate in nursing, Grand Canyon University's (GCU) Bachelor of Science in Nursing (RN to BSN) program online can prepare you to take your career to the next level. Earn your BSN to sharpen your skills, broaden your understanding of best practice and prepare to take on more responsibilities as a leader in a hospital or healthcare organization. GCU's RN-BSN is tailored to meet the needs of the RN adult learner and to maximize the strengths that the working RN already possesses. Our university counselors provide an upfront time to completion, and our free cost estimator tools provide you with a general estimation of the cost of attendance.
Bachelor of Science in Psychology (Online)
The psychology major offers a broad array of courses that increase our understanding of human actions in the past and present. With this understanding comes the responsibility to improve our relationships with others and to help others to achieve similar understanding and insight into their own behaviors. The program helps prepare the student for entry-level positions in agencies and organizations which seek to help those in need. It should be understood that the requirements for employment in many positions in the helping professions include advanced education beyond the bachelor's degree.
Bachelor of Science in Psychology - Forensic Psychology (Online)
Grand Canyon University's Bachelor of Science in Psychology with an Emphasis in Forensic Psychology degree program provides an overview of the intersection of psychology and the criminal justice system. This program is for students interested in working with people who need assistance coping with the aftereffects of a violent crime. Students who are curious about the cause of crime as well as the role of the legal system and psychology may find this career path desireable. The American Psychological Association defines forensic psychology as "the application of clinical specialists to the legal arena" or applying clinical psychology skills (assessment, treatment and evaluation) to forensic settings.
Bachelor of Science in Psychology - Performance and Sports Psychology (Online)
The Bachelor of Science in Psychology with an Emphasis in Performance and Sport Psychology is designed for students with a keen interest in sports, performing arts, health and fitness or mental health fields. Channel your passion for human behavior while learning to guide performance on the highest level with this sports psychology degree. Psychological wellness is a key component of competitive greatness. Study the foundations of behavioral health and learn how to bring out the best in diverse populations.
Bachelor of Science in Public Health (Online)
Grand Canyon University's Bachelor of Science in Public Health program prepares students to work as public health practitioners in various health-related settings, including community-based organizations, government agencies, behavioral health agencies, primary care centers and global or domestic nonprofit organizations. The coursework is designed to ensure that graduates of this program can carry out day-to-day tasks of public health organizations, such as data collection and analysis, program planning, community outreach, communications and program support. The BS in Public Health program also integrates the National Commission for Health Education Credentialing (NCHEC) Seven Areas of Responsibility for Health Education Specialists, addressing fundamental skills in community assessment, program planning, implementation and evaluation.
Bachelor of Science in Risk Management (Online)
In GCU's Bachelor of Science in Risk Management program, students will learn about the risk management process including identifying and analyzing risk, understanding how risk impacts businesses/individuals and what mitigation techniques such as insurance may be available to protect assets. In addition, students will study | 0.234987 | -0.252767 | 1.997665 | 0.345067 | [
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organizational and enterprise risk management. Finally, students will learn about the accounting cycle and the construction of financial statements, examine the market forces of supply and demand under different market structures, discuss the basic functions of the monetary system, analyze the macro economy in terms of long-run economic productivity and growth and in terms short-run fluctuations, build financial models to help analyze a company and more.
Bachelor of Science in Software Development (Online)
As computers only continue to become more necessary in our society, those equipped with skills to spur this growth onward will be prepared for high-demand careers. Graduates of GCU's Bachelor of Science (BS) in Software Development degree are able to work with a variety of programming languages, mobile and web technologies, web application frameworks and today's cloud platforms. With project-driven courses and industry expert faculty, students will develop the skills needed to enter this field after graduation.
Bachelor of Science in Sports and Entertainment Management (Online)
Grand Canyon University's Bachelor of Science in Sports and Entertainment Management degree is uniquely positioned within the Colangelo College of Business. Program coursework emphasizes key skills specific to sports business including sales, marketing, revenue generation, athlete and talent business and management as well as event operations and the program's business-based curriculum prepares students for an array of potential career path opportunities in the multi-billion-dollar sports and entertainment industry. This program aligns with GCU's Christian mission, aiming to develop graduates committed to upholding ethical and moral principles within the fields of entertainment and athletics.
Bachelor of Science in Supply and Logistics Management (Online)
Grand Canyon University's Bachelor of Science in Supply Chain and Logistics Management degree program addresses servant leadership, innovation, entrepreneurialism, and business skills. In addition to foundational courses in business, students complete courses in lean operations, business process management, and enterprise optimization. Other courses focus on procurement, logistics and systems aspects of global supply chains. Students also learn about Lean Six Sigma Green Belt certification.
Bachelor of Social Work (Online)
Created by the College of Humanities and Social Sciences, this online social work degree program prepares students to assist in improving the wellbeing of individuals, families and communities while working with diverse groups of people in a variety of settings.
Master of Science in Computer Science (Bridge) (Online)
Grand Canyon University's Bridge (Master of Science in Computer Science) program is designed to be an accelerated pathway into GCU's Master of Science in Computer Science degree program for those interested in engineering, information technology, science and more specifically, a career in computer science. A bachelor's degree, which can be in an unrelated field, is required.
Master of Science in Data Science (Bridge) (Online)
Grand Canyon University's Bridge (Master of Science in Data Science) program provides an accelerated pathway for those interested in pursuing a rewarding career in data science. This bridge program serves as an introduction to data science and is available for anyone with a bachelor's degree in any field who wishes to enter into GCU's Master of Science in Data Science offered by the College of Science, Engineering and Technology. Students will gain the foundation necessary to begin their academic journey toward a future in data science and business analytics.
Master of Science in Information Technology Management - Bridge (Online)
Grand Canyon University's Bridge (Master of Science in Information Technology Management) degree program is designed for students without a technology-related degree who aspire to gain the leadership skills and business management knowledge in information technology. Offered within the Colangelo College of Business, this IT management degree program will provide you with a strong information technology foundation necessary to move forward toward earning your master's degree. You will develop the training and participate in hands-on activities to gain the skills and understanding that you can apply during the MS in IT Management program.
Master of Science in Nursing - Health Care Informatics - Bridge (Online)
Transform your healthcare career at the intersection of two fast-growing fields: healthcare and information technology. The Bridge (Master of Science in Nursing with an Emphasis in Health Care Informatics) is for experienced RN's who desire to improve healthcare using cutting-edge technology and data-driven insights. Students in this nursing bridge pathway, offered through the College of Nursing and Health Care Professions, have earned a bachelor' | 0.287378 | -0.225908 | 1.875925 | 0.480338 | [
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s degree outside of nursing and desire to gain foundational knowledge in preparation for the MSN degree.
Master of Science in Nursing - Health Care Quality & Patient Safety - Bridge (Online)
Grand Canyon University's Master of Science in Nursing (MSN) in Patient Safety and Health Care Quality bridge pathway was created to support registered nurses with bachelor's degrees in fields other than nursing. Created by the College of Nursing and Health Care professions, this bridge pathway helps students gain foundational knowledge and allows them to prepare for their MSN degree in a timely and convenient manner. Lead the Future of Patient Quality and Safety
Master of Science in Nursing - Leadership in Health Care Systems - Bridge (Online)
Join leaders and change-makers on the forefront of nursing advancement. The Bridge (Master of Science in Nursing with an Emphasis in Nursing Leadership in Health Care Systems) pathway, offered by Grand Canyon University's College of Nursing and Health Care Professions, helps experienced RN's with a bachelor's degree outside of nursing make the transition to master's degree studies. Upon completing the six courses in the nursing bridge pathway, you will enter the MSN with an emphasis in nursing leadership. In this MSN bridge pathway, you will develop your skills in critical thinking, effective communication, leadership and administration of resources in preparation for a management role in today's complex healthcare environments.
Master of Science in Nursing - Nursing Education - Bridge (Online)
Master of Science in Nursing: Nursing Education (Bridge). This pathway is designed for registered nurses with a bachelor's degree in health sciences or other related fields who desire to pursue a graduate degree in nursing. Courses in the bridge pathway are a blending of RN to BSN core courses that are considered essential requisites to graduate study. Upon completion of bridge course from the BSN core,students enter one of five emphases offered for the Master of Science in Nursing. The Nursing Education emphasis prepares students for a variety of roles in nursing education,either as clinical educators or academic educators in both traditional and nontraditional settings in nursing education. Graduates are prepared to assess educational needs,plan pathways to meet those needs,and evaluate the outcomes of their pathways. The curriculum includes an in-depth examination of nursing theory,roles and practice; research,ethics,policy and finance in the health care system; and pathophysiology,pharmacology and health assessment for nurse educators. In addition,curriculum design and technology for nursing education is explored. A capstone course provides an opportunity for students to develop an evidence-based project proposal that addresses a problem or issue in professional practice. Graduates of this program will possess the knowledge and skills needed to successfully teach nursing in a wide variety of settings.
Master of Science in Nursing - Public Health - Bridge (Online)
Offered by the College of Nursing and Health Care Professions, the Bridge (Master of Science in Nursing with an Emphasis in Public Health) pathway allows experienced RN's with a non-nursing bachelor's degree to embark on the road to earning an MSN public health degree. Upon completing foundational nursing courses from the Bachelor of Science in Nursing, you will then enter the MSN pathway which offers an intensive introduction to key skills and concepts in public health analysis, planning and implementation. As an MSN bridge pathway student, you will experience a six-course program of study in nursing upper division level subject matter.
Academic Profile of Incoming Freshman
Application FeeN/A
Early Decision Application DeadlineN/A
Regular Application DeadlineN/A
High School RankNot applicable
High School RecordNot applicable
Completion of College-Prep ProgramNot applicable
Admissions Test Scores (ACT/SAT)Not applicable
TOEFL (Test of English as Foreign Language)Not applicable
Books & SuppliesN/A
Average Cost Per Credit HourN/A
Estimated Financial Aid
Receiving Any Financial AidN/A
of Students Receiving Financial Aid Average Financial
Aid Amount
Federal GrantsN/AN/A
P | 0.03296 | -0.598806 | 1.997665 | 0.493345 | [
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Library Laundries in Napa
https://napavalleyregister.com/news/local/wash-the-clothes-read-a-book-library-laundries-come-to/article_fc6a1440-8d2a-5992-acf7-f9066f81a7ed.html
Wash the clothes, read a book: Library laundries come to Napa
JENNIFER HUFFMAN [email protected] | Dec 10, 2019 Updated 21 hrs ago
A handful of local laundromats are now offering something extra to go along with the spin and dry cycles.
Books. Library books.
The library has come to the laundromat.
Thanks to a California Library Literacy Services Family Literacy grant, the Napa library has installed children's "libraries" inside two Napa laundromats. And more library laundries are on the way.
"It's the perfect place" to promote reading, said Robin Rafael, literacy and volunteer services supervisor at the Napa County Library.
Forbes Correlates Literacy and Workforce Readiness
https://www.forbes.com/sites/jimcowen/2019/11/12/an-urgent-call-for-improved-reading-and-literacy-instruction/
An alarm sounded last week when results from the Nation's Report Card showed faltering reading levels for America's young people. The release from the National Association of Educational Progress (NAEP) revealed a decrease in student achievement in 17 states in fourth grade reading and 31 states in eighth grade reading in the last two years. With all of the "breaking news" in our day to day lives, this headline is worthy of real reflection. Our students and our schools need and deserve our help.
Beyond just the common sense nature of needing to be able to read and write well, according to the 2016 Why Reading Matters report from Business Roundtable, "93 percent of CEOs rated reading and writing as very or somewhat important for current job openings in their companies, and 42 percent of CEOs reported problems in finding qualified applicants with strong enough skills in those areas." On top of that, the Georgetown Public Policy Institute reports that we will have 55 million job openings in the economy throughout 2020. Sixty-five percent of them will require a postsecondary education and training beyond high school and the fastest growing occupations (health care, community services and STEM) have the highest demand. How will we possibly be able to meet our workforce needs if our young people continue to struggle to read at a proficient level?
I am proud that organizations representing education and civil rights communities have quickly come together to address this alarming news. This week, more than ten leading education groups have issued an urgent call to action to focus on five critical areas:
Embracing the science of reading – Instruction in early reading, in particular, has the most solid research behind it of anything in education. We know what works, now let's make sure it's in practice in every school and classroom.
Pushing for use of high-quality, standards- aligned curriculum – States like Mississippi, which had positive NAEP reading results, illustrate what's possible when reading strategies are implemented patiently and effectively.
Advancing the capacity of teachers in teaching literacy by way of teacher preparation programs – A 2018 report from the National Council on Teacher Quality (NCTQ), a nonprofit organization that prioritizes teacher effectiveness, found that only 37% of elementary and special education teacher preparation programs included evidence-based methods of reading instruction.
Assisting non-profits, business and community leaders in their efforts to strengthen reading strategies for their students – Equipping parents with resources and best practice so that they can become effective partners in this work. (Emphasis mine)
Increasing the federal investment in literacy for local and state evidenced-based, comprehensive literacy efforts – While primary responsibility for teaching children to read and write lies at the state and local level, this national challenge warrants federal support.
It is encouraging that movement is already underway by state education leaders to focus attention on this area. For example, the Council of Chief State School Officers announced that in early 2020, they will hold a "Literacy Summit" with national and state leaders to | 0.819496 | 1.014102 | 2.159984 | 0.06672 | [
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] | 1,332,919 | 0 | CommonCrawl | 21 | 17 |
Spiritual Odysseys
Kabbalah 101
Home » Spirituality » Kabbalah 101
https://www.aish.com/sp/k/48950986.html
Holy Sparks #5: Why Not Antarctica?
by Rabbi Pinchas Winston
Some spots on earth are merely storage halls of Holy Sparks, until the time comes for each spark to be released.
Let's face it. If being sent to the "four corners of the earth" is to be taken literally, there are still plenty of places the Jewish people have not yet moved to, willingly or unwillingly. Not that anyone else has moved there either, but still, for Jews uncomfortable with the Middle East situation and not 100 percent secure in America (or England or France or South Africa or Argentina), the Great White North, or the Great White South, might be a last ditch option. Not to mention the Sahara Desert...
Why not? It might be far easier to weather the cold and polar bears, and avoid all the social, political and religions tensions. Maybe God has one more final exile in mind for the Jewish people, a much colder one, after which time we'll have gone to all those corners and be ready to come home, once and for all.
When God created the world, He separated out a large portion of [holy sparks] to make all that was to come into existence throughout the six days of creation. What remained after that (of the sparks from the world of Tohu), the Almighty left behind for man to rectify through his actions, throughout the generations until the arrival of Moshiach.
They were set aside and given place in all the deserts throughout the world, from where they ascend and are constantly undergoing rectification. What this means is that they are drawn from the portions of the desert to places of settlement, where they become revealed and rectified as necessary.
They represent the new and additional strength of each yield from the land, which results in straw for animals and wheat from which man makes his bread, or anything that is "quarried" from the ground. As it is written, 'For silver has a source and there is a place where they refine gold' (Job 28:1). All of it is the result of the rejuvenation which flows into the places of settlement from the places of desert. (Sefer HaKlalim, Klal 19:2)
At first reading, this sounds like a difficult concept to understand. However, as we will see, not only does it makes perfect sense, but it helps us better appreciate a miracle of life that we may otherwise take for granted.
HIDDEN SYSTEM OF LIFE
Why are there new crops every year? Why is there plenty in some years, and famine in others? Why does the ground at times produce nothing at all, let alone food that we can eat and enjoy? We may know why this occurs on a physical level, but what is the spiritual mechanism?
Holy Sparks, of course.
Like a time capsule, each Holy Spark of history has its own moment in time to be realized and used throughout the course of six millennia, before returning to its source Above. Who determines when that moment is and where it is to be used? God Himself, Who moves the sparks to wherever He wants, in order to help a person, nation, and the world to fulfill its potential and role on the stage of world history. Only God can be so precise.
God has an elaborate system for moving sparks around, so they're available when and where we need them.
God has a whole system for doing it, and right under our very noses, too. The life that we take for granted in a world that we are used to, is really a very sophisticated and elaborate system for moving sparks around, so they can be available when and where we need them.
For example, a simple wind moving north from the Sahara Desert toward a big city may be carrying dust filled with Holy Sparks, just waiting to land on the soil of some community to bring life to the ground, which in turn can bring life to animals. This in turn brings life to people who can use the Holy Sparks for a Godly purpose -- thereby expending them and sending them on their way back to the source.
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ation in the world could not make a difference.
Similarly, what gives a vitamin its power is the sparks contained within, which are invisible to the scientist and doctor. Why else would one entity be stronger or weaker than another in terms of accomplishing a specific task?
It all comes down to the sparks inside -- the nature of a specific spark and its potential. Why one place in the world is inhabitable and why one is not depends on the sparks there, and their potential to maintain life on any level. How that particular place responds to external conditions and human activity will depend on its spiritual make-up -- that is, the sparks designated to be there.
Think of it as a spiritual gene pool. Each spark has a different potential to create, vis a vis the other sparks. Appreciating the vastness of life and its closeness to unlimited potential, is another way of appreciating the vast spiritual pool of Holy Sparks. From there we can begin to appreciate a little of the wisdom God used to make creation, and with which He continues to maintain creation every moment of the day, all throughout history.
INHABITABLE BY SPARKS, NOT MAN
However, the question we came to answer was why there are no longer any "corners" of the world where the Jewish people can go. The answer is that the deserts, whether they are frozen or made from sand, seem to be places not meant to be inhabited by humans.
Of course, if we had to, we could build some kind of enclosed city in the Arctic Circle, or in the middle of the Sahara Desert. Knowing human ingenuity, we'd figure out some way to sustain life within it.
When Adam left the Garden of Eden, he designated specific places on earth to be inhabitable.
However, such areas in the world are not "natural" human habitats, and at this late time in history it seems unlikely that they will become so. More than likely they are wastelands because God ordained them as places that man will not live en masse, so that they can remain as storage halls of Holy Sparks, until the time comes for each individual spark to be released and its potential realized.
In fact, the Talmud says that when Adam left the Garden of Eden, he designated specific places on earth that would be inhabitable (Brachot 31a).
Perhaps Holy Sparks are the reason why there are no more "corners" of the earth for the Jewish people to flee to, as they have continually done throughout their long, arduous journey toward the end of history and the Final Redemption.
In any case, the Jews are in the final exile of history: Golut Edom -- the Exile of Edom, Eisav's descendants. It began with the Romans back at the destruction of the Second Temple, and in the 2,000 years since, we've been everywhere Edom has made his home -- whether in Europe, Russia, or America.
We've been nearly everywhere mankind has ever settled.
Now it's time to go home.
The Jewish year is 5762. According to the Zohar's calculation, with which the great kabbalist Rabbi Shlomo Elyashiv seems to concur, "Resurrection of the Dead" is destined to begin no later than 28 years from now. By that time, the Zohar also explains, all the exiled Jews remaining at the end of history will have to be gathered in, and Moshiach will have already rid the world of evil.
By this time, the sparks of history will have to already be expended and returned to their sources Above.
Rome wasn't built in a day, and it probably would take a lot longer to build a thriving community in inhospitable Antarctica. No matter how we look at it, if life for the Jewish people takes a turn for the worst, and the "Lavans" of today no longer look at us with the same smile as they did yesterday, then we need to think about whether we have perhaps exhausted the sparks.
If our work is finished, then staying around may defeat our purpose. And in the end, defeat us, too.
See Holy Sparks Part 1: Exile and Redemption
See Holy Sparks #2: The Contractor Who Couldn't Say Goodbye
See Holy Sparks Part 3: Soul Food
See Holy Sparks Part 4: The Purpose of Creation
Article 5 of 5 in the series Holy Sparks
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parks #3: Soul Food
Exile and Redemption Part 1: Holy Sparks
Holy Sparks #4: The Purpose of Creation
Rabbi Pinchas Winston
Rabbi Pinchas Winston, an Aish alumni, is the author of over 50 books on Torah philosophy, the weekly parshah sheet PERCEPTIONS, and his site Thirtysix.org. He hosts the "Ask the Rabbi" radio show for Arutz Sheva (Israel National Radio), and is in the process of translating into English "Sha'ar HaGilgulim," the Arizal's work on reincarnation (completed chapters can be viewed at: http://shaarhagilgulim.thirtysix.org/Shaar_HaGilgulim/Introduction.html).
(5) Yehudith Shraga, July 22, 2012 3:24 PM
The pool of Holy sparks needs some expalantion
Who determines when and ...where it( holy spark) to be used?-G-d Himself.It is a little bit difficult to navigate in the spiritual ocean, but anyway some order should be made,there is nothing to be afraid that only G-d knows when and where the holy spark is going to meet you,because G-d never gives as situations which we may not deal with.There is Devine order within the Sparks according to their difficulty of release,there are Sparks of Shoret,Alef, Bet,Gimel Ovi'ut and the most difficult-Ovi'ut Dalet which is our Antarticas,Desorts, High mountains and oceans of potential,but to release the spark from there we use a special tech. called Hakarat haRah=acknowledgement of Evil, we start with out personal inner work of course,and our incorrected parts are shown to us by the Creator according to our willingness to correct them,Schar Mitzva-Mitzva is the point=the more evil we willing to acknowledge within ourselves the more we are shown,first are shown very easy parts to our ego to ackowledge=ovi'ut Shoresh,if we progress in our spiritual work of correction we are shown the parts belonging to the ovi'ut Alef and so on. Shoresh, Alef, Bet belong to Tzimtzum Bet which is combination of Justice and Mercy,but then games turn into the grown-up life,the Ovi'ut Gimmel and especially Dalet are for supermen/women only,which were our ForFathers, ForMothers,Prophits and Sages, it's a real test of our preference of spiritual over material(here it is the place to remind that Kabbalah's notion- Body means the amount of ego the person is given,and not his meat, blood body which is the dress for his ego!).All the knowledge of the structure of the spiritual worlds is in Torah,Gemmara,Midrashim, our Sages' books which are mentioned in the article and of course the book of razi'el, the book of Creation, the book of Zohar,the books of Ar'i teachings and Baal haSulam's Talmud Esser HaSeffirot,which should be studied, and IMPLEMENTED, because we promissed to the Creator to Do and to Hear.
(4) Dennis Gentry, September 18, 2002 12:00 AM
WOW!!!!!!!!
I am truly finding out how little I do know, keep up the good work Aish
(3) David L, February 16, 2002 12:00 AM
According to my knowledge, it's quite the contrary. A region with an abundance of sparks, draws the Jewish Nation towards it, for the purpose of uplifting the sparks. Whereas there are regions that are spiritual wastelands of sparks, and therefore don't warrant our noble civilization.
In addition, after God gave us the Torah, we have the ability to gather sparks from remote locations too, since the world was created with the torah, and therefore it contains the spiritual soul of every time and place.
(2) anita vanput, February 13, 2002 12:00 AM
intelligent,the last great adventure-thank you
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Last season I got exhausted because I didn't expect to play that much golf and I didn't train for it. "That's probably been the lesson I've learned through all this, there are days I just have to shut it down and not do anything and relax because most of my career has been spent on, "In order to get better, I have to work at it".
This defeat was due to our mentality more than anything else, our mental approach. "I spoke with him also and I want to him be ready for the match today, for the next matches like all the players in the team". "We have the best defensive midfielder in world football and you go and buy Jorginho", Simon added. "It's always hard for a player but we'll see what happens".
She has only reached this stage of a Grand Slam once since she was stabbed in the hand by an intruder in her home in 2016. U.S. tennis star Danielle Collins has surprised many with a run to the Australian Open quarters, after beating three seeded opponents including number two Angelique Kerber in Melbourne on Sunday.
The settlement today came as the search continues to find the plane carrying new Cardiff striker Sala . Sala rejected the opportunity to sign mega-deal with a Chinese club and instead penned in a deal with Neil Warnock's side, joining from French outfit Nantes for £15 million (R269 million).
Translation: they won't cut Lance Stephenson, Ivica Zubac, Michael Beasley, or anyone now guaranteed. The deal won't be official until Tuesday, but Anthony will never play for the Bulls , but ESPN's Adrian Wojnarowski reported Monday that the team may wait until after the February 7 trade deadline to release him.
In ODIs, Kohli led India to nine victories, with four losses and one tied match. "They have improved a lot and this is because their board had the foresight to set up the MRF pace academy years back and get Dennis Lillee as their first coach to help groom their pacers", Abbas said.
They were very typical of what Sean Payton likes to call, short passes that substitute for a running game thrown in the hopes that you'll loosen up the front for a rushing attempt. This time, New Orleans (14-4) couldn't hang on to the lead or overcome that officiating mistake. The New Orleans Advocate had a similarly infuriated headline: "They Blew The Call".
Nadal really only experienced hard moments midway through the second set when he had to save two break points in a 10-minute game. Federer , 37, was the oldest player left in the field, while Tsitsipas was the youngest remaining in the fourth round.
Open title, has two major women's doubles titles but lost the only final she reached at Melbourne Park in 2006. The Russian, who retired in 2003, will be inducted to the International Tennis Hall of Fame this July in Newport. Meanwhile Pierce, who is a former world No. 3 in singles and doubles, said "To be recognized alongside some of the most accomplished, inspiring people in our sport is a huge honor".
Lee was four strokes back of the co-leaders. Ji shook off a birdie-bogey start and answered a bogey at six with a birdie two holes later, then birdied 13 and 16 and closed with two pars to stay in the second-place pack. At 32 years, eight months and seven days, Ji is now the oldest South Korean victor on the LPGA Tour, breaking the previous record of 32 years, seven months and 18 days set by Hall of Famer Pak Se-ri in May 2010.
If that was a wake-up call to his players as they chase Liverpool and prepare for a Champions League showdown with Schalke and an FA Cup fourth-round tie against Burnley and, surely, a Carabao Cup final, it would be a stretch to say Guardiola is alarmed.
Louis in a recent interview. Machado, who has played shortstop and third base in the majors, is a career.282 hitter with 175 home runs and 513 RBIs. He's the kind of star player and middle-of-the-order bat who can play shortstop for the Bucs, so there is a baseball fit here. He wrote an Instagram post about Bryant and D | -0.700843 | -0.945741 | -1.375536 | -1.211853 | [
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empster, saying that players who criticize other cities are "stupid players and losers".
The date and time of the game will be announced along with the schedule this spring. The contest is a home game for the Buccaneers, meaning the Panthers will have eight regular season games at Bank of America Stadium per usual. Mexico City hosts the other global game and will feature the Kansas City Chiefs at Los Angeles Chargers . In addition to London, the Panthers will play road games at Arizona, San Francisco, Green Bay, Indianapolis, Houston, Atlanta and New Orleans.
An obvious pass interference penalty against the Rams went uncalled late in regulation and they beat the Saints in overtime. After the game, Payton told reporters that he was told by the NFL's head of officials that the officials in the game "blew the call", according to ESPN's Mike Triplett.
That means Ronaldo will not be able to drive into a court auto park away from the cameras, as he had requested. Soccer star Cristiano Ronaldo is expected to plead guilty to four counts of tax evasion on Tuesday in Madrid but likely will avoid jail.
It may not be the magic that the 2017 season had for Foles, but he was undoubtedly one of the primary reasons that the Eagles could reach a playoff berth in a down year. So now that Philadelphia's offseason has officially started, let's take a look at some potential trade destinations for Foles in case this is what the Eagles end up deciding to do, even if some of the teams seem like absolute long shots.
Another social media user Richard Ings said people should not underplay how serious the incident was. Barty took advantage of an off day from Sharapova on serve; the Russian double-faulted on 10 occasions throughout the match, gifting the No.
He will now play as a stand-in striker for Luiz Suarez at Barcelona who is on a superb goal scoring form for the Catalan giants this campaign. "Born in Berlin on 6 March 1987, Kevin-Prince Boateng is a player who will surely adapt quickly to the team", said Barcelona on their official website.
Farrell has had a procedure to fix ligament damage and is expected to return to action in seven to 10 days, a time frame that would enable him to play against opening Six Nations opponents Ireland in Dublin on February 2. Wasps showed some resilience in the second half with Michele Campagnaro went close and eventually set up Nathan Hughes over the line in the 53rd minute and Lima Sopoaga improved it.
Dates and locations of those games will be announced at a later date. The schedule for the National Football League games in London and Mexico City for 2019 was released on Monday, with the Rams set to play in the United Kingdom for the fourth time as they take on the Cincinnati Bengals.
The 18-year-old was in tears after her drubbing and Williams comforted her at the net, telling her "you're gonna make it, don't cry". "I tend to keep a lot of things bottled up". She said no, but later signalled a thumbs-up. She said it released the tension and her fightback began. Seeded four, Osaka has defeated No. 13 seed Anastasija Sevastova 4-6 6-3 6-4.
Agree or disagree with Solskjær's comments about Rashford? 'Definitely, ' said Solskjaer without hesitation. He can be absolutely top-class. Rashford scored a superb goal to put his side 2-0 up in the Premier League contest at Old Trafford and caused the Seagulls plenty of other problems as United's ideal start under Solskjaer extended to a seventh game in all competitions.
There was a minor wobble when Williams allowed Halep to bring up three break points in the opening game, conceding with a double fault. Simona Halep , right, congratulates Serena Williams after the American repelled a fight-back from the world No1.
The Nets acquired Faried from the Denver Nuggets in an offseason trade that largely amounted to a dump of his remaining $13.8-million salary. Faried, 29, spent his first seven seasons in the National Basketball Association with Denver, which selected him with the 22nd overall pick in the 2011 draft out of More | -0.861095 | -0.960066 | -1.69003 | -1.392648 | [
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TEAM NEWS: Andy Ellis among four changes for York City Knights
TEAM NEWS: Andy Ellis among four changes for York City Knights ahead of Rochdale Hornets' visit
By Joe Richardson @joerichpress Sports reporter
HANDY ANDY: Andy Ellis - who was instrumental in York City Knights' League One-winning campaign last year - returns to James Ford's side for the rest of the season. Picture: Gordon Clayton
EXPERIENCED hooker Andy Ellis is set to feature for York City Knights in their Betfred Championship clash against Rochdale Hornets on Sunday.
Head coach James Ford is looking forward to seeing the 34-year-old in action again after bringing back the willing Ellis to plug the gap left by pack injuries for the rest of the season.
Ellis hung his boots up for the third time in 2018, having been instrumental in the Knights' League One-winning campaign and making his 350th career appearance in the final-day victory over Whitehaven that clinched the title.
But with all three hookers out of action - Kriss Brining ineligible to play through concussion and Will Jubb and Harry Carter both struggling with knocks - Ford has called on the reliable veteran.
Ford said: "I've spoken to Andy a couple of times and he's keen to see how the boys are going.
"Tuesday, we weren't sure whether Kriss Brining would complete his concussion protocol, and he hasn't.
"I said, 'What do you think about helping us out?' and Andy being Andy, he put his hand up."
He added that Ellis has been keeping himself match-fit by playing at a lower level.
"He looks sharp in training and we're looking forward to including him," Ford said.
"I am delighted that Andy has agreed to help us through this hectic schedule. We trained on Tuesday without a recognised nine and, although none are long-term issues, it was certainly a concern.
"It's a great fit as he knows our systems, terminology and philosophies. He's a vastly experienced player who will be a great boost.
"He's really well thought of on and off the field and I'm looking forward to coaching him again."
Ellis began his career with Hull KR in 2003 aged 19, scoring 18 tries in 50 appearances, and lifting the old Northern Rail Cup in 2005 with victory over Castleford.
He then starred for Barrow during a brief golden era for the Cumbrians, taking them up to the Championship and was named that tier's Man of Steel after leading them to the 2009 title and the National Rail Cup final before spending nearly three years in Super League with Harlequins and Wakefield.
Following a 2012 loan move turned permanent, Ellis helped Featherstone Rovers top the Championship that year, and again 12 months later.
He briefly retired at the end of 2015 to concentrate on teaching but returned and went on to play 24 times in 2016, including in all seven matches in the Middle Eights. He played 120 times in all for Rovers, notching 29 tries.
He hung up his boots again but came back to join the Knights in March 2017.
Ford makes three other changes to the squad that faced Toulouse. Jack Blagborough (oblique) is injured, Harry Aldous has been recalled by Newcastle Thunder and Joe Porter does not make the squad.
This will also be the last game of Junior Vaivai and Will Oakes' month-long loan from Hull KR. The Knights are in talks with Rovers and Ford is "hopeful" of getting them back.
York: (from) Marsh, Bass, Salter, Vaivai, Whiteley, Harris, Robinson, Teanby, Jubb, Horne, Scott, Jordan-Roberts, Spears, Ellis, Dixon, Dupree, Stock, Hey, Oakes.
Connor Robinson pens new deal with York City Knights
York Cricket Club aiming to build on their "dream come true" in the County Championship
York City 2 Hartlepool United 1: Vote for your man of the match
Boxer Davey turns professional with Warren
Scott extends Knights loan until end of 2 | -0.437181 | -0.360205 | -1.128252 | -1.232664 | [
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Represent the best quality and lowest priced jet ski parts and accessories online! ..
New Sea-Doo 4Tec Bellow with sealed ball bearing. This fits all Sea-Doo 4Tec models. This is the seal that keeps the oil in the PTO while the drive shaft is inserted into the crank shaft. OEM#: 420832648 Applications 2002 Sea Doo GTX 4-TEC Ball Bearing with Bellow 2003 Sea Doo GTX 4 TEC Ba..
New Ritco Products Ignition coils. Sold as a set of 3. Out performing competitors aftermarket ignition coils. Ritco Products coils Perform higher allowing your spark plug to burn fuel at a better rate. RItco products offers a 1 Year replacement warranty.New Seadoo Ignition coil. Ritco Products..
New Sea-Doo Exhaust removal and installer tool. This tool helps by easily accessing the 15mm nut that is underneath the down pipe that is fastened to the exhaust manifold. Applications 1997 Sea-Doo GSX Limited Exhaust Wrench 1998 Sea-Doo GSX Limited Exhaust Wrench 1998 Se..
Sea-Doo Cam and Crank shaft locking tool kit. This kit is used so you can properly time your seadoo engine. This is used for all seadoo four-stroke engines. OEM 529035821 OEM 529035829 Applications ALL GTI 4TEC RXP Supercharged 215 , 255 & 260 RXT 215, 255 GTX NA GTX S..
New Sea-Doo drive line center bearing. This driveline is manufactured of T6061 aircraft aluminum complete with grease fitting, new center installed bearing and outer seals. OEM 272-00-024 Applications SP 1990-1995 GT 1990-1991 XP 1991-1995 All SPI 1993-1995 SPX 1993-1995 GTS 1992-1995 G..
New Sea-Doo early style fuel knob. Replaces Part no 275-000-134 This is a direct oem replacement part for early model seadoos Applications 1990-1991 GT 1992-1996 GTS 1992-1995 GTX 1989-1995 SP / SPI / SPX 1992 XP ..
New Sea-Doo Late style fuel knob. Replaces Part no 275-000-263 This is a direct oem replacement part for all late model Sea-Doo selector switches. ..
This fuel selector Switch has 5/16 inlets and outlet. Manufactured of casted machined aluminum with durable rubber seals. Including a new plastic selector knob for Reserve, On and Off. ..
New Sea-Doo inline fuel filter with replacement O-ring. OEM 275500090 OEM 275500089 Applications 1991 Sea-Doo GT Fuel Filter with O-Ring 1992 Sea-Doo GTS Fuel Filter with O-Ring 1992 Sea-Doo GTX Fuel Filter with O-Ring 1992 Sea-Doo SP Fuel Filter with O-Ring 1992 Sea-Do..
Jetskijunky offers an easy affordable fix for that damaged jet pump housing, this housing includes new aluminum casted housing, and impeller shaft, wear ring and a new jet pump rebuild kit. Pre-assembled so this way it saves you plenty of time to get back on the water. Just install your impell..
Jetski Junky offers an easy affordable fix for that damaged jet pump housing, this housing is an exact OEM replacement for your Seadoo. Includes impeller shaft, wear ring and a new jet pump rebuild kit. Pre-assembled so this way it saves you plenty of time to | -2.045865 | -1.11585 | -1.740755 | -1.512312 | [
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This Little Art
Kate 21stC, Andre Gide, Dorothy Bussy, getting published, Helen Lowe-Porter, Kate Briggs, poetry, Roland Barthes, Thomas Mann, translation February 15, 2018 February 14, 2018
Kate Briggs' meditation on the art of translation meanders around her experience of translating some lectures given by Roland Barthes in (I think) the 1970s, her deep interest in the translations by Helen Lowe-Porter of Thomas Mann, and the relationship between Andre Gide and his besotted translator Dorothy Bussy. I'm not interested in Barthes, but I am interested in the process of translation, so the sheer volume of Barthes in this book is rather off-putting to the non-Barthesian. Some things that she quotes him saying are relevant, but his work is mostly there as the kicking-off point for Briggs' own thoughts on what translation is, and why and how she does it. Her empathy for Lowe-Porter stems from the kicking that (female) translator got by the later (male) critics for her inaccuracies and general inadequacies, long after she was dead and unable to defend her methods. Bussy is also an object of sympathy, even though her uncritical adoration of Gide, for all her life, seems a little like self-imposed slavery. Translation is a thankless task, and gets vicious responses from interested parties: no-one knows better than translators that translation is exhausting and never final.
Speaking dragonese is one way of thinking about translation: should you put on a dragon's accent when reading aloud a dragon's words in a story? (Especially if the typeface for their words is different from that of other characters' dialogue.) So is thinking of translation as a speculative version of what was really said. It is a new version, a parallel version, a different but closely related version. 'There's something from the outset speculative and, I would say, of the novelistic about the translator's project'. 'The translator as an unknown instrument: a tool to be used, a service provider, engaged in undressing and carefully redressing the literary work of art for the purposes of a new market … just let me change the garment of your art'.
Briggs is particularly good on translation as a literary art: 'an ever-renewable demonstration of the literary value of the novel … literature, that quality we call the literary, simply cannot do without translation as a means of repeatedly affirming it (and when the words of a translation matter in turn, when we feel, in a translation, that it must indeed be these necessary words in this necessary order, the translation has become literature too'.
She describes a clever exercise in challenging prejudices about translation: giving translation students two texts, one a translation of the other, but with the labels transposed. The original text was thus critiqued dismissively as a poor effort of translation from the translation itself, and when the exchange was revealed, much backtracking ensued. 'The categories of "original" and "translation" act more like placeholders … names for the positions we put writing in'.
I most enjoyed the stories from Briggs' own translating life, particularly the episode of trying to identify the haiku that Barthes had translated. He was working from Japanese into French, and she was working from her own mental translation of French into English. She found the original copy of the book that Barthes had used, in the Bibliothèque Nationale, but the haiku – mentioning the flower that Barthes calls a 'carnation' – could not be found, until – in exasperation – Briggs let the book's pages fall open, and there it was, a 'pink'. From Japanese to French to English, the flower had moved its name from one culture to another.
Translations leave things out as well as add them. Abbé Prévost's French edition of Pamela reduced the original seven volumes in English to four in French (and French is a wordier language than English). The first English translation of Simone de Beauvoir's The Second Sex lost 15% of the original. Paul Legault 'translated' Emily Dickinson's poems into single lines, also in English.
Translation is like a winter plum, to be watered by a pale, slender monk; or a peony, to be bathed by a handsome maiden in full costume. Or so Barthes translated from The Book of Te | 2.123427 | 0.79296 | 1.046574 | 0.906963 | [
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2015 Eugene Brevdo <[email protected]>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_SPECIAL_FUNCTIONS_H
#define EIGEN_SPECIAL_FUNCTIONS_H
namespace Eigen {
namespace internal {
/****************************************************************************
* Implementation of lgamma *
****************************************************************************/
template<typename Scalar>
struct lgamma_impl
{
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(const Scalar&)
{
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
template<typename Scalar>
struct lgamma_retval
{
typedef Scalar type;
};
#ifdef EIGEN_HAS_C99_MATH
template<>
struct lgamma_impl<float>
{
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE double run(const float& x) { return ::lgammaf(x); }
};
template<>
struct lgamma_impl<double>
{
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE double run(const double& x) { return ::lgamma(x); }
};
#endif
/****************************************************************************
* Implementation of erf *
****************************************************************************/
template<typename Scalar>
struct erf_impl
{
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(const Scalar&)
{
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
template<typename Scalar>
struct erf_retval
{
typedef Scalar type;
};
#ifdef EIGEN_HAS_C99_MATH
template<>
struct erf_impl<float>
{
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE float run(const float& x) { return ::erff(x); }
};
template<>
struct erf_impl<double>
{
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE double run(const double& x) { return ::erf(x); }
};
#endif // EIGEN_HAS_C99_MATH
/***************************************************************************
* Implementation of erfc *
****************************************************************************/
template<typename Scalar>
struct erfc_impl
{
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(const Scalar&)
{
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
template<typename Scalar>
struct erfc_retval
{
typedef Scalar type;
};
#ifdef EIGEN_HAS_C99_MATH
template<>
struct erfc_impl<float>
{
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE float run(const float x) { return | -0.946015 | -0.83114 | -0.005332 | 1.289364 | [
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Peanut Butter Wolf - Seattle 9/14/13
People Music presents
Adv Ticket Sales have ended.
Tickets available at the door for $15. Set Times: Shapey 9-10 Zac Hendrix 10-11 The New Law 11-12 Peanut Butter Wolf 12-1:30
FACEBOOK EVENT POST RE-BAR 1114 Howell St Seattle, WA 98101
@pbwolf (@stonesthrow) Seattle 9/14 @rebarseattle w/ @thenewlaw @ZacHendrix <Info> http://t.co/3tfDi0pzhQ <tickets> http://t.co/ORv9UnPfkg
— People Music (@People_Music) August 13, 2013
https://www.stonesthrow.com
Peanut Butter Wolf, is a world renowned DJ, hip-hop producer, and founder of the hip-hop record label Stones Throw Records.
Among the true talents in the late-'90s new skool of old-school hip-hop, Peanut Butter Wolf began DJing as a teenager and became quite an entrepreneur at his San Jose, CA, high school, selling mix tapes of his turntable work. He debuted on wax in 1989 with "You Can't Swing This" on All Good Vinyl, recorded with a smooth MC named Lyrical Prophecy. By the end of the year, PBW began working with MC Charizma, and the two gelled quickly, perfecting their skills at block parties and shows with the likes of the Pharcyde, House of Pain, and Nas. Just after the duo gained a record deal with Hollywood Basic in 1992, Charizma was shot and killed.
Unsure of where to turn without the talents of his MC, Peanut Butter Wolf began issuing strictly instrumental work, including his first release, Peanut Butter Breaks on Heyday Records. Tracks followed for the Bomb label (on the excellent Return of the DJ compilation) as well as Om Records. Though the MC had ruled the world of hip-hop since the mid-'80s, selected DJ crews began to get exposure in the dance underground by the late '90s, including DJ Shadow, the Invisibl Skratch Piklz, and the X Men (whose name was later changed to the X-Ecutioners, for copyright reasons). Peanut Butter Wolf joined the elite as well, recording for dance labels 2 Kool (the Lunar Props EP) and Ninja Tune (remixing the Herbaliser) as well as contributing production work for fellow old-schooler Kool Keith.
With the foundation of his Stones Throw label, PBW began developing tracks recorded earlier with MC Charizma and released work by another Bay Area crew, Fanatik. His debut production LP, My Vinyl Weighs a Ton, appeared in 1999. The year 2002 brought two excellent compilations, The Best of Peanut Butter Wolf and the Stones Throw 7" collection Jukebox 45's. In 2003 he put out Badmeaninggood, Vol. 3 and the next year he produced a compilation CD/DVD of his label's artists, Stones Throw 101. Since then he has focused more of his energy in building up Stones Throw, though he still performs the occasional DJ gig in Europe, Australia, Canada, Japan, and the U.S. Peanut Butter Wolf is also a member of the L.A. hip-hop orchestra Breakestra. To celebrate the decade his label had been in existence, Chrome Children was released in collaboration with Adult Swim in 2006, and Stones Throw: Ten Years in early 2007.
The New Law Seattle based producers Adam Straney and Justin Neff have created a unique style of instrumental music featuring organic and synthetic hip hop beats combined with evolving soundscapes, dreamlike sax, dusty samples, soaring synths, and pulsing drum machines that can only be defined as The New Law.
The New Law by THE NEW LAW
Zac Hendrix is a dj, producer, and all around funky cat, as | -0.150234 | -0.038787 | -1.329883 | -1.011548 | [
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It's the first Sunday in March. I am hoping winter is behind us, but I have lived in England long enough to know better. We will have many more cold, dark, windy days before spring rescues us. Still, today the sun encourages a promising blue sky, with just the right amount of fat white clouds.
I saw two, tiny, baby lambs, in the fields with their mums, a few hours ago. Earlier, we treated ourselves to a walk in the countryside, along a river, with plenty of company from other winter weary ramblers. At least half the people had dogs with them, fueling my dog loving instincts. I have been wanting a pair of terriers, of indeterminate breed, for some time now. Richard, though is a cat person, not a dog lover, so we make do with a garden full of birds, visiting the nine feeders he keeps going year round.
One of my favourite farm shops is, coincidentally, hollerin' distance from where we stroll. Poking along, avoiding the worst of the mud, I was thinking about what I might buy at Castle Farm. This is the place I first discovered the sensual pleasures of lavender ice cream. It has ruined me for all other dairy confections.
I will definitely buy lavender ice cream. And beef. They raise their own beef at Castle Farm. It's the only place I've seen in England, selling beef with the bone intact. Everywhere else I shop, sells boneless beef, though I suppose our local butcher would keep the bone intact, were I to ask. With luck, in a few more months, we'll be sitting in the garden with a couple of t-bones on the grill.
By the time we return home from our country pursuits, with beef and lavender ice cream happily in tow, I'm famished. Unable to contain myself, I indulge in two generous scoops of ice cream, before settling on what to make for lunch. The newly purchased beef will keep for a couple of days, but three chicken breasts, in the back of the fridge, command my attention.
Much as I adore winter food, I am a bit tired of soups and chowders; casseroles and stews. Even curries, which I don't make very often, fail to inspire my imagination. Thinking back to the chicken satay and stir fried vegetables, I recently had for lunch in Tunbridge Wells, I decide to re-create them, while bird watching from the kitchen window.
A pretty pheasant stops by to help herself to the banquet of bird food Richard provides. I smile at the irony of observing food eating food, while I make food. Quiet as I try to be, creeping into the garden with my camera, I manage to get a single picture of the pheasant before she makes a run for it. No pheasant under glass for dinner tonight.Pulling out the ingredients I need to make chicken satay, I spy lime leaves. Lime leaves are new to me. When I spotted them last week, in the freezer section of the grocery store, I tossed them into the cart, knowing they would come in handy. Cooking is usually great fun for me, as long as I keep surprising myself with new ingredients and recipe ideas.
Lime leaves feed into my sense of play. They are the perfect "What If?" ingredient. What if I take these lime leaves and mix them up with some coconut milk and peanut butter? What if I add a bit of soy sauce and lime juice to the mixture? Would I not have the makings for a killer sauce?
What if I cut chicken breast into long strips and poach it in the peanut buttery, lime-leaf-infused-sauce for half an hour or so? What if I add some dried oriental noodles to the pot at the end? How would it taste?
Mmmmmm. It tastes like a meal you might get in a really good Thai restaurant. I used to go to such a restaurant in San Rafael, California. I loved the place. I never knew lime leaves could add so much flavour to a dish. I never knew the secret of lime leaves until today.
After making this dish, I wished we were eating this with friends. It is so good and easy to whip up | 1.707046 | -0.241128 | -0.182868 | -1.174133 | [
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\section{Introduction}
The following frequently used feature of spectra of periodic second order ordinary differential operators in $L^2(\R)$ is well known and follows from the standard Floquet theory \cite{MagnusWinkl,Eastham}: all edges of the spectral bands of such an operator on the whole axis can be found by considering the spectra of periodic and antiperiodic boundary value problems. In the multi-dimensional case, the role of the periodic and anti-periodic problems is played by symmetry points of the Brillouin zone. Thus, until recently, there had been a rather wide spread belief that these were the only possible locations of the spectral edges. This belief was in most cases confirmed by numerics done for crystalline or photonic crystal media. It was, however, the experience of numerical analysts that in general this claim should be incorrect. This was shown analytically to be indeed the case in \cite{HKSW}. However, after that, claims have been encountered that in the case of a single periodicity (i.e., with respect to the lattice $\Z$), the spectral edges do occur at the periodic and anti-periodic spectra only. In this work we show that even in the simplest case of quantum graphs this is not true. Namely, it is shown in Theorem \ref{T:main} that this is indeed true if the graph consists of a $1D$ chain of finite graphs connected by single edges, while if the connections are formed by two or more edges, the spectral edges can already occur away from the periodic and anti-periodic spectra. Thus, one has to be careful drawing conclusions about $1D$-like spectral behavior when the medium is $1D$-periodic.
We will be considering a graph $G$ that is an infinite ``chain'' of identical copies $\G_j$ of a graph $\G$, consecutive copies being connected by $m$ edges. E.g., Fig. \ref{F:chain} shows such a chain graph with $m=3$ connecting edges. Here the internal structure of the repeated copies $\G_j$ is not important and thus is not revealed.
\begin{figure}[ht!]
\begin{center}
\setlength{\unitlength}{2cm}
\begin{picture}(5,1)
\put(0.0,0.0){\includegraphics[width=10cm]{chain.eps}}
\put(0.8,0.35){$\Gamma_{j-1}$}
\put(2.4,0.35){$\Gamma_j$}
\put(4.0,0.35){$\Gamma_{j+1}$}
\end{picture}
\end{center}
\caption{A $\Z$-periodic ``chain'' graph $G$ with three connecting edges.}\label{F:chain}
\end{figure}
The graph $G$ is equipped with the natural action by the shifts of the group of integers $\Z$. We will denote this action as $(n,x)\in \Z\times G \mapsto \tau_n x \in G$.
We will assume $G$ to be a quantum graph (see, e.g., the survey \cite{KuAGA} and references therein). In other words, each edge $e$ is assigned a finite length $0<l_e<\infty$ in such a way that it is invariant with respect to the $\Z$-action. Correspondingly, each edge $e$ is provided with a ($\Z$-periodic on the whole $\G$) coordinate function $x_e$ (or just $x$), which identifies $e$ with the segment $[0,l_e]$. Moreover, an operator (Hamiltonian) $\H$ is defined, which acts as the (negative) second derivative $-\rmd^2/\rmd x^2$ on each edge, accompanied with the conditions at the vertices, which require the functions from its domain to be continuous, and their outgoing derivatives along edges to sum to zero at each vertex (the so called Neumann or Kirchhoff vertex conditions \cite{KuAGA}). We define the domain of $\H$ as consisting of all functions $f$ on $G$ such that
\begin{enumerate}
\item $f\in H^2(e)$ for each edge $e$, where $H^2(e)$ is the standard Sobolev space of order $2$ on the segment $e$ (e.g., \cite{KuAGA}),
\item
\begin{equation}\label{E:h2cond}
\sum\limits_e \|f\|^2_{H^ | 0.333604 | -0.034311 | 0.740957 | 1.846058 | [
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] | 1,332,984 | 0 | ArXiv | -1 | -1 |
2(e)}<\infty,
\end{equation}
\item $f$ satisfies at each vertex $v$ the vertex conditions specified above:
\begin{equation}\label{E:Kirchhoff}
f \mbox{ is continuous at }v \mbox{ and }
\sum\limits_e \frac{\rmd f}{\rmd x_e}(v)=0,
\end{equation}
where the sum is taken over all edges adjacent to the vertex $v$ and derivatives are taken in the directions away from $v$.
\end{enumerate}
It is known that this unbounded operator $\H$ in $L^2(G)$ is self-adjoint \cite{KuAGA}.
Due to the $\Z$-periodicity of $\H$, the graph version of the standard Floquet theory applies (see, e.g., \cite{Eastham,MagnusWinkl,RS,Ku93,Ku01} for the Floquet theory for differential equations, and references in \cite{KuAGA} for its graph version). It implies the following features that we will need to use. Let $k\in [-\pi,\pi]$ be a {\bf quasi-momentum} (the name comes from the solid state physics, e.g. \cite{AM}). We consider the Floquet Hamiltonian operator $\H (k)$ that is defined similarly to $\H$, with the exception that the second condition (\ref{E:h2cond}) is replaced by the automorphicity (Floquet, cyclic) condition
\begin{equation}\label{E:h2cond_k}
f(\tau_nx)=\rme^{\rmi kn}f(x) \mbox{ for any }(n,x)\in \Z\times G.
\end{equation}
Notice that when $k=0$, condition (\ref{E:h2cond_k}) reduces to the $\Z$-periodicity of $f$, while $k=\pm \pi$ produces the anti-periodic condition. These two cases play an important role in our considerations.
The spectrum of $\H (k)$ is discrete and, numbered in non-decreasing order (counting multiplicity), consists of the eigenvalues $\lambda_j(k), \quad j=1,2,\dots$. The functions $\lambda_j(k)$ are called the {\bf band functions}. The closed interval $I_j$, which is the image of the $j$-th band function, $I_j:=\lambda_j([-\pi,\pi])$, is called the $j$-th {\bf band}. The multiple-valued function
$$
\lambda(k):=\{\lambda_j(k)\}
$$
is said to be the {\bf dispersion relation} and its graph $B$ is called the {\bf Bloch variety}. The level sets of the dispersion relation are often called {\bf Fermi surfaces} (although in physics this term is used for one specific level \cite{AM}).
The following result is well known both in ODE and graph situations:
\begin{theorem}\label{T:Floquet}(see \cite{RS,Eastham,KuAGA,Ku01,Ku93,Titchmarsh} and references therein) \indent
\begin{enumerate}
\item The band functions $\lambda_j(k)$ are continuous and piece-wise analytic.
\item The Bloch variety is an analytic set, i.e. it can be described as the set of all zeros of an analytic function $\phi(k,\lambda)$.
\item The spectrum of the operator $\H$ in $L^2(G)$ consists of the union of the bands:
\begin{equation}\label{spectr}
\sigma(\H)=\bigcup\limits_{j}I_j=\bigcup\limits_{k\in[-\pi,\pi]} \sigma(\H(k)).
\end{equation}
\end{enumerate}
\end{theorem}
Our goal is to consider in the $\Z$-periodic quantum graph case the validity of the analog of the following classical theorem:
\begin{theorem}\label{T:ODE}(see \cite{RS,Eastham,Ku01,Ku93,MagnusWinkl,Titchmarsh} and references therein) In the ODE case (i.e., when the graph $G$ is just the straight line), the endpoints of the bands $I_j | 0.538713 | 0.217273 | 1.120125 | 2.543225 | [
0.06069622033643107,
1.0167296697775658
] | [
-0.3906458445744054,
0.8251909621348936
] | [
0.5063550181702113,
1.7338956744561975
] | [
2.3247096393691384,
2.7617398444356227
] | 1,024 | [
512,
512
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=\lambda_j([-\pi,\pi])$ (i.e., the extrema of the band functions) are attained at the points $k=0,k=\pm \pi$. In other words, the spectra of the periodic and anti-periodic problems provide the ends of the bands of the spectrum.
\end{theorem}
\section{The main result}
We will now discuss the validity of an analog of Theorem \ref{T:ODE} in the quantum graph case.
\begin{theorem}\label{T:main} Let $G$ be a $\Z$-periodic ``chain'' graph $G$ with $m$ connecting edges (Fig. \ref{F:chain}) and $\H$ be the corresponding Hamiltonian operator in $L^2(G)$. Then
\begin{enumerate}
\item If $m=1$, the endpoints of the bands $I_j=\lambda_j([-\pi,\pi])$ (i.e., the extrema of the band functions) are attained at the points $k=0,k=\pm \pi$ (although, they might be attained at some other points as well). In other words, the spectra of the periodic and anti-periodic problems provide the ends of the bands of the spectrum.
\item If $m>1$, this is not always true.
\end{enumerate}
\end{theorem}
The first statement of the theorem essentially claims that in terms of the spectral band edges, the chain quantum graphs with single connecting edges behave similarly to ODEs.
We start with the following crucial auxiliary statement:
\begin{lemma}\label{L:dimension}\indent
\begin{enumerate}
\item If some value $\lambda$ is attained by the band functions $\lambda_j(k)$ at more than two points $k$ in the segment $(-\pi,\pi]$, then there is a constant branch $\lambda(k)\equiv \lambda$ for all $k$, and thus this value is attained at all points of the segment.
\item The set $D$ of all such values $\lambda$ is discrete (possibly empty).
\item If $\lambda\notin D$, then in a neighborhood of this value all band functions are strictly monotonic on $[0,\pi]$.
\end{enumerate}
\end{lemma}
{\bf Proof of the Lemma.}\phantom{M}
Figure \ref{F:chain1} shows a link of a chain graph with $m=1$.
\begin{figure}[ht!]
\begin{center}
\setlength{\unitlength}{2cm}
\begin{picture}(3,1)
\put(0.0,0.0){\includegraphics[width=6cm]{chain1.eps}}
\put(0.4,0.6){$x_1$}
\put(2.5,0.6){$x_2$}
\put(1.4,0.4){$\Gamma$}
\end{picture}
\end{center}
\caption{One link of a chain graph with $m=1$, where the two marked points are related by the unit shift: $x_2=\tau_1 x_1$. The internal structure of the link $\G$ is irrelevant and not shown.}\label{F:chain1}
\end{figure}
Suppose that a value $\lambda$ is taken by band functions $\lambda_j(k)$ at more than two points $k\in [-\pi,\pi]$. This means that the space of (bounded) solutions of the equation $(\H-\lambda \I)u=0$ on $G$ is more than two-dimensional. Hence, there exists a non-trivial solution $u$ that vanishes with its first derivative at a point $x_1$ on the connecting edge (see Fig. \ref{F:chain1}). Since the equation is of the second order on each edge, we conclude that $u$ vanishes on the whole edge containing $x_1$. Suppose that $u$ is not identically equal to zero on the part of the graph to the right of $x_1$ (the case when this happens to the left can be treated in a similar manner). Re-defining $u$ as being equal to zero everywhere to the left of $x_1$, one has a legitimate bounded solution, which we will denote by $u$ again. Then the linear hull of the right shifts of $u$ is infinitely dimensional. However, the Floquet theory (e.g., \cite{Ku_graphs2}) implies that then the ``Fermi surface'' of all | -0.126949 | -0.549396 | 0.694037 | 1.570312 | [
0.06617492205242612,
-0.3200735489252243
] | [
-0.2837677011274815,
-0.8150248183448087
] | [
0.9324435104347686,
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1.9397068396677115,
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points $k\in [0,\pi]$ where $\lambda_j(k)=\lambda$ for some $j$ must also be infinite. Then the analytic structure (see Theorem \ref{T:Floquet}) of the Bloch variety
$$
B=\{(k,\lambda)\mid \lambda_j(k)=\lambda \mbox{ for some }j\}
$$
implies that it contains the flat branch $\lambda_j(k)\equiv\lambda$ and that this can occur only at a discrete set $D$ of values of $\lambda$. This proves the first two statements of the Lemma.
Consider now a value $\lambda\notin D$. Due to the invariance of the operator $\H$ with respect to the complex conjugation (which corresponds to the time reversal symmetry of the system), the band functions $\lambda_j(k)$ are even: $\lambda_j(-k)=\lambda_j(k)$. This, together with the first statement of the Lemma and the assumption that $\lambda\notin D$ implies that all values near $\lambda$ are attained by the (continuous) function $\lambda_j(k)$ only once on $[0,\pi]$. Thus, this function is monotonic there, which proves the last claim of the Lemma.
\qed
\noindent
{\bf Proof of Theorem \ref{T:main}.}\phantom{M} Suppose that $\lambda$ is a band edge, i.e. an extremum of a band function $\lambda_j(k)$. If $\lambda\in D$, where $D$ is the set defined in the Lemma, then the statement (i) of the Lemma claims that this value $\lambda$ is attained at all values of quasi-momentum $k$, in particular for $k=\pi$ and $k=0$ that correspond to the anti-periodic and periodic problems.
Let now $\lambda\notin D$. Then the statement (iii) of the Lemma implies that the corresponding value of $k$ cannot be in the interior $(0,\pi)$ of the segment $[0,\pi]$. Thus, either $k=0$ and $\lambda$ belongs to the spectrum of the periodic problem, or $k=\pi$ and $\lambda$ belongs to the spectrum of the anti-periodic problem. This proves the first statement of the theorem.
Let us now construct an example with $m=2$, where some of the band edges are attained not on the spectra of the periodic and anti-periodic problems. We start with the $\Z^2$-periodic graph considered in \cite[Fig. 1]{HKSW}. Its fundamental domain $W$ (with the dotted line boundary) is shown in Figure \ref{fig:quantumfd} below.
\begin{figure}[ht!]
\begin{center}
\setlength{\unitlength}{4cm}
\begin{picture}(1,1)
\put(0.0,0.0){\includegraphics[width=4cm]{quantumfd.eps}}
\put(0.26,0.41){$4$} \put(0.53,0.41){$5$} \put(0.74,0.41){$2$}
\put(0.55,0.257){$3$} \put(0.55,0.71){$1$} \put(0.7,0.74){$W$}
\put(0.05,0.41){$9$} \put(0.55,0.02){$8$} \put(0.98,0.41){$7$}
\put(0.55,0.98){$6$}
\end{picture}
\caption{The graph $\Gamma$ with fundamental region $W$.}
\label{fig:quantumfd}
\end{center}
\end{figure}
One imagines the whole graph as obtained by tiling the plane with the $\Z^2$-shifted copies of $W$. The lengths of all edges are assumed to be equal to $1$. The two basic period vectors are $v_1$ connecting vertices $4$ to $7$ and $v_2$ connecting $3$ to $6$.
It was shown in \cite{HKSW} that some of the band functions of the discrete Laplace operator on this graph attain their extremal values strictly inside the reduced Brillouin zone $[0,\pi]^2$. It was then shown that this implies the same feature for the quantum graph operator $\H$.
We obtain our example by folding this graph along the period vector | 0.822236 | -0.07197 | 0.795486 | 2.054167 | [
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$v:=v_1+v_2$. In other words, the points $x$ and $y$ such that $y=v+x$ are identified. This leads to the $\Z$-periodic graph $G$ shown in Fig. \ref{F:Zgraph}.
\begin{figure}[ht!]
\begin{center}
\setlength{\unitlength}{5cm}
\begin{picture}(1,1)
\put(0.0,0.0){\includegraphics[width=5cm]{Zgraph.eps}}
\put(0.55,0.55){$W$}
\end{picture}
\end{center}
\caption{The $\Z$-periodic graph obtained by folding the graph of Fig. \ref{fig:quantumfd}.}\label{F:Zgraph}
\end{figure}
This graph is clearly $\Z$-periodic and has $m=2$ (the two skew edges connecting the adjacent copies of the fundamental domain $W$).
Let us consider the version of the discrete Laplace-Beltrami operator $\Delta$ on functions defined on the vertices of $G$, which adds the values at all neighboring vertices,
weighted by the reciprocals of the square roots of the degrees
(valences):
$$
(\Delta f)(v):=\frac{1}{\sqrt{d_v}}\sum\limits_{u\sim v}\frac{1}{\sqrt{d_u}}f(u),
$$
where $u$ and $v$ are vertices, $d_u$ is the degree of $v$, and the sum is taken over all vertices $u$ adjacent to $v$ (in the paper \cite{HKSW} this operator was denoted $L$).
For instance, for a function $f$ defined on the vertices of the graph in Fig. \ref{fig:quantumfd} one has
$$
(\Delta f)(5)=\frac1{\sqrt{3}}\left(\frac1{\sqrt{3}}f(1)+\frac12f(3)+
\frac1{\sqrt{3}}f(4)\right),$$
and
$$ (\Delta f)(3)=\frac12\left(\frac1{\sqrt{3}}f(2)+\frac1{\sqrt{3}}f(4)+
\frac1{\sqrt{3}}f(5)+\frac1{\sqrt{3}}f(8)\right), \mbox{{\it etc}.}
$$
Then direct calculation shows that for any quasi-momentum $k$, the Floquet operator $\Delta(k)$ is given by the following $5\times 5$ matrix:
\begin{equation}\label{E:matrix}
\Delta(k)= \frac1{\sqrt{3}}\left(
\begin{array}{ccccc}
0 & 3^{-1/2} & \rme^{\rmi k}/2 & 0 & 3^{-1/2} \\
3^{-1/2} & 0 & 1/2 & \rme^{\rmi k}/3^{1/2} & 0 \\
\rme^{-\rmi k}/2 & 1/2 & 0 & 1/2 & 1/2 \\
0 & \rme^{-\rmi k}/3^{1/2} & 1/2 & 0 & 3^{-1/2} \\
3^{-1/2} & 0 & 1/2 & 3^{-1/2} & 0 \\
\end{array}
\right)
\end{equation}
The plot of the five eigenvalues of $\Delta(k)$ for $k=(l-1)\pi/20,l=1\ldots21$ is shown in Fig. \ref{F:plot}.
\begin{figure}[ht!]
\begin{center}
\scalebox{.6}{\includegraphics{plot.eps}}
\end{center}
\caption{The graphs over $[0,\pi]$ of the five band functions of $\Delta$.}\label{F:plot}
\end{figure}
One sees that the minimum of the third branch, as well as the maximum of the second one are attained inside the segment, and thus not at the points $k=0$ or $k=\pi$. This numerical observation can be confirmed, as in \cite{HKSW}, by a somewhat tedious explicit calculation of the dispersion relation and elementary analysis of its behavior. Again, as in \cite{HKSW}, one shows that this fact about the band functions of the discrete operator $\ | 0.478447 | 0.091033 | 0.681356 | 2.043762 | [
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] | 1,332,984 | 4,096 | ArXiv | -1 | -1 |
@interface CBL_TestSubModel : NSObject <CBLJSONEncoding>
- (instancetype) initWithFirstName: (NSString*)firstName lastName: (NSString*)lastName;
@property (readonly, copy, nonatomic) NSString *firstName, *lastName;
@end
@interface CBL_TTestMutableSubModel : CBL_TestSubModel
@property (copy, nonatomic) NSString *firstName, *lastName;
@end
@implementation CBL_TestSubModel
{
@protected
NSString* _firstName, *_lastName;
}
@synthesize firstName=_firstName, lastName=_lastName;
- (instancetype) initWithFirstName: (NSString*)first lastName: (NSString*)last {
self = [super init];
if (self) {
_firstName = first;
_lastName = last;
}
return self;
}
- (id) initWithJSON:(id)jsonObject {
return [self initWithFirstName: jsonObject[@"first"]
lastName: jsonObject[@"last"]];
}
- (id) encodeAsJSON {
return @{@"first": self.firstName, @"last": self.lastName};
}
- (BOOL) isEqual:(id)object {
return [self.firstName isEqual: [object firstName]] && [self.lastName isEqual: [object lastName]];
}
@end
@implementation CBL_TTestMutableSubModel
{
CBLOnMutateBlock _onMutate;
}
- (void) setOnMutate:(CBLOnMutateBlock)onMutate {
_onMutate = onMutate;
}
- (void) setFirstName:(NSString *)firstName {
_firstName = [firstName copy];
if (_onMutate) _onMutate();
}
- (void) setLastName:(NSString *)lastName {
_lastName = [lastName copy];
if (_onMutate) _onMutate();
}
@end
@interface CBL_TestModel : CBLModel
@property int number;
@property unsigned int uInt;
@property NSInteger nsInt;
@property NSUInteger nsUInt;
@property int8_t sInt8;
@property uint8_t uInt8;
@property int16_t sInt16;
@property uint16_t uInt16;
@property int32_t sInt32;
@property uint32_t uInt32;
@property int64_t sInt64;
@property uint64_t uInt64;
@property bool boolean;
@property BOOL boolObjC;
@property float floaty;
@property double doubly;
@property NSString* str;
@property NSData* data;
@property NSDate* date;
@property NSDecimalNumber* decimal;
@property CBL_TestModel* other;
@property NSArray* strings;
@property NSArray* dates;
@property NSArray* others;
@property NSDictionary* dict;
@property CBL_TestSubModel* subModel;
@property CBL_TTestMutableSubModel* mutableSubModel;
@property NSArray* subModels;
@property int Capitalized;
@property unsigned reloadCount;
@end
@implementation CBL_TestModel
@dynamic number, uInt, sInt16, uInt16, sInt8, uInt8, nsInt, nsUInt, sInt32, uInt32;
@dynamic sInt64, uInt64, boolean, boolObjC, floaty, doubly, dict;
@dynamic str, data, date, decimal, other, strings, dates, others, Capitalized;
@dynamic subModel, subModels, mutableSubModel;
@synthesize reloadCount;
- (void) didLoadFromDocument {
self.reloadCount++;
Log(@"reloadCount = %u",self.reloadCount);
}
+ (Class) othersItemClass {
return [CBL_TestModel class];
}
+ (Class) datesItemClass {
return [NSDate class];
}
| -1.418553 | -0.945741 | -0.481828 | 0.688448 | [
-1.3500694715322923,
-1.4870370144321683
] | [
-0.9475313333617279,
-0.9439500761991084
] | [
-0.2722709766227593,
-0.6913848060675097
] | [
0.7534819973443964,
0.6234134839317521
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512,
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] | 1,332,988 | 0 | Github | -1 | -1 |
+ (Class) subModelsItemClass {
return [CBL_TestSubModel class];
}
@end
#pragma mark - MODELS:
#define TEST_PROPERTY(PROPERTY, VALUE) \
model.PROPERTY = VALUE; \
CAssertEq(model.PROPERTY, VALUE); \
CAssertEqual([model getValueOfProperty: @""#PROPERTY], @(VALUE));
TestCase(API_ModelDynamicProperties) {
NSArray* strings = @[@"fee", @"fie", @"foe", @"fum"];
NSData* data = [@"ASCII" dataUsingEncoding: NSUTF8StringEncoding];
CBLDatabase* db = createEmptyDB();
CBL_TestModel* model = [[CBL_TestModel alloc] initWithNewDocumentInDatabase: db];
TEST_PROPERTY(number, 1337);
TEST_PROPERTY(number, INT_MAX);
TEST_PROPERTY(number, INT_MIN);
TEST_PROPERTY(uInt, UINT_MAX);
TEST_PROPERTY(uInt, 0u);
TEST_PROPERTY(sInt64, INT64_MAX);
TEST_PROPERTY(sInt64, INT64_MIN);
TEST_PROPERTY(uInt64, UINT64_MAX);
TEST_PROPERTY(uInt64, 0u);
TEST_PROPERTY(sInt32, INT32_MAX);
TEST_PROPERTY(sInt32, INT32_MIN);
TEST_PROPERTY(uInt32, UINT32_MAX);
TEST_PROPERTY(uInt32, 0u);
TEST_PROPERTY(sInt16, INT16_MAX);
TEST_PROPERTY(sInt16, INT16_MIN);
TEST_PROPERTY(uInt16, USHRT_MAX);
TEST_PROPERTY(uInt16, 0u);
TEST_PROPERTY(sInt8, INT8_MAX);
TEST_PROPERTY(sInt8, INT8_MIN);
TEST_PROPERTY(uInt8, UCHAR_MAX);
TEST_PROPERTY(uInt8, 0u);
TEST_PROPERTY(nsInt, NSIntegerMax);
TEST_PROPERTY(nsInt, NSIntegerMin);
TEST_PROPERTY(nsUInt, NSUIntegerMax);
TEST_PROPERTY(nsUInt, 0u);
TEST_PROPERTY(boolean, false);
CAssertEq([model getValueOfProperty: @"boolean"], (id)kCFBooleanFalse);
TEST_PROPERTY(boolean, true);
CAssertEq([model getValueOfProperty: @"boolean"], (id)kCFBooleanTrue);
TEST_PROPERTY(boolObjC, NO);
TEST_PROPERTY(boolObjC, YES);
TEST_PROPERTY(floaty, 0.0f);
TEST_PROPERTY(floaty, (float)M_PI);
TEST_PROPERTY(doubly, 0.0f);
TEST_PROPERTY(doubly, M_PI);
TEST_PROPERTY(Capitalized, 12345);
model.str = @"LEET";
model.strings = strings;
model.data = data;
CAssertEqual(model.str, @"LEET");
CAssertEqual(model.strings, strings);
CAssertEqual(model.data, data);
Log(@"Model: %@", [CBLJSON stringWithJSONObject: model.propertiesToSave options: 0 error: NULL]);
[db _close];
}
TestCase(API_ModelEncodableProperties) {
CBLDatabase* db = createEmptyDB();
CBL_TestModel* model = [[CBL_TestModel alloc] initWithNewDocumentInDatabase: db];
| -1.739057 | -1.031691 | -0.628395 | 0.130454 | [
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-1.6897489779239847
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-1.2161256205581856,
-0.8472561328083836
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-0.6989935291436625,
-0.5577962358085977
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] | 1,332,988 | 1,024 | Github | -1 | -1 |
CBL_TestSubModel* name = [[CBL_TestSubModel alloc] initWithFirstName: @"Jens" lastName: @"Alfke"];
model.subModel = name;
AssertEq(model.subModel, name);
AssertEq([model getValueOfProperty: @"subModel"], name);
NSMutableDictionary* props = [model.propertiesToSave mutableCopy];
[props removeObjectForKey: @"_id"];
CAssertEqual(props, (@{@"subModel": @{@"first": @"Jens",
@"last": @"Alfke"}}));
CBLDocument* doc2 = [db createDocument];
CAssert([doc2 putProperties: props error: NULL]);
CBL_TestModel* model2 = [[CBL_TestModel alloc] initWithDocument: doc2];
CAssertEqual(model2.subModel, name);
// Now test array of encodable objects:
CBL_TestSubModel* name2 = [[CBL_TestSubModel alloc] initWithFirstName: @"Naomi" lastName: @"Pearl"];
model.subModel = nil;
NSArray* subModels = @[name, name2];
model.subModels = subModels;
AssertEqual(model.subModels, subModels);
AssertEq([model getValueOfProperty: @"subModels"], subModels);
CBL_TTestMutableSubModel* name3 = [[CBL_TTestMutableSubModel alloc] initWithFirstName: @"Jed" lastName: @"Clampett"];
model.mutableSubModel = name3;
AssertEq(model.mutableSubModel, name3);
AssertEq([model getValueOfProperty: @"mutableSubModel"], name3);
props = [model.propertiesToSave mutableCopy];
[props removeObjectForKey: @"_id"];
CAssertEqual(props, (@{@"subModels": @[@{@"first": @"Jens", @"last": @"Alfke"},
@{@"first": @"Naomi", @"last": @"Pearl"}],
@"mutableSubModel": @{@"first": @"Jed", @"last": @"Clampett"}}));
CBLDocument* doc3 = [db createDocument];
CAssert([doc3 putProperties: props error: NULL]);
CBL_TestModel* model3 = [[CBL_TestModel alloc] initWithDocument: doc3];
Assert(!model3.needsSave);
CAssertEqual(model3.subModels, subModels);
CAssertEqual(model3.mutableSubModel, name3);
// Mutate the submodel in place and make sure the model's serialization changed:
name3 = model3.mutableSubModel;
AssertEqual(name3.lastName, @"Clampett");
Assert(!model3.needsSave);
name3.lastName = @"Pookie";
Assert(model3.needsSave);
props = [model3.propertiesToSave mutableCopy];
CAssertEqual(props, (@{@"subModels": @[@{@"first": @"Jens", @"last": @"Alfke"},
@{@"first": @"Naomi", @"last": @"Pearl"}],
@"mutableSubModel": @{@"first": @"Jed", @"last": @"Pookie"},
@"_id": doc3.documentID,
@"_rev": doc3.currentRevisionID}));
[db _close];
}
TestCase(API_ModelEncodablePropertiesNilValue) { // See #247
RequireTestCase(API_ModelEncodableProperties);
CBLDatabase* db = createEmptyDB();
CBL_TestModel* emptyModel = [[CBL_TestModel alloc] initWithNewDocumentInDatabase: db];
AssertNil(emptyModel.mutableSubModel);
NSString* documentID = [[emptyModel document] documentID];
emptyModel = nil;
CBLDocument *document = [db documentWithID:documentID];
emptyModel = [[CBL_TestModel alloc] initWithDocument:document];
AssertNil(emptyModel.mutableSubModel);
[db _close];
}
TestCase(API_ModelDeleteProperty) {
NSArray* strings = @[@"fee", @"f | -1.328155 | -0.742504 | -0.171455 | 0.246215 | [
-1.2240593320644062,
-1.4322499972722178
] | [
-0.6055212743315715,
-0.8794874472719585
] | [
-0.09219786382047616,
-0.250712927906993
] | [
0.33726275442393483,
0.15516683564623296
] | 1,024 | [
512,
512
] | [
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353,
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29961,
5507,
1725,
29872,
613,
12490,
29888
] | 1,332,988 | 2,048 | Github | -1 | -1 |
ie", @"foe", @"fum"];
NSData* data = [@"ASCII" dataUsingEncoding: NSUTF8StringEncoding];
CBLDatabase* db = createEmptyDB();
CBL_TestModel* model = [[CBL_TestModel alloc] initWithNewDocumentInDatabase: db];
model.number = 1337;
model.str = @"LEET";
model.strings = strings;
model.data = data;
CAssertEqual(model.str, @"LEET");
CAssertEqual(model.strings, strings);
CAssertEqual(model.data, data);
model.data = nil;
CAssertEqual(model.data, nil);
model.data = data;
NSError* error;
CAssert([model save: &error], @"Failed to save: %@", error);
CAssertEqual(model.data, data);
model.data = nil;
CAssertEqual(model.data, nil); // Tests issue CouchCocoa #73
[db _close];
}
TestCase(API_SaveModel) {
NSDate* date = [NSDate dateWithTimeIntervalSinceReferenceDate: 392773252];
NSArray* dates = @[date, [NSDate dateWithTimeIntervalSinceReferenceDate: 392837521]];
NSDecimalNumber* decimal = [NSDecimalNumber decimalNumberWithString: @"12345.6789"];
CBLDatabase* db = createEmptyDB();
NSString* modelID, *model2ID, *model3ID;
{
CBL_TestModel* model = [[CBL_TestModel alloc] initWithNewDocumentInDatabase: db];
CAssert(model != nil);
CAssert(model.isNew);
CAssert(!model.needsSave);
modelID = model.document.documentID;
CAssertEqual(model.propertiesToSave, @{@"_id": modelID});
// Create and populate a TestModel:
model.number = 1337;
model.str = @"LEET";
model.strings = @[@"fee", @"fie", @"foe", @"fum"];
model.data = [@"ASCII" dataUsingEncoding: NSUTF8StringEncoding];
model.date = date;
model.dates = dates;
model.decimal = decimal;
CAssert(model.isNew);
CAssert(model.needsSave);
CAssertEqual(model.propertiesToSave, (@{@"_id": model.document.documentID,
@"number": @(1337),
@"str": @"LEET",
@"strings": @[@"fee", @"fie", @"foe", @"fum"],
@"data": @"QVNDSUk=",
@"date": @"2013-06-12T23:40:52.000Z",
@"dates": @[@"2013-06-12T23:40:52.000Z",
@"2013-06-13T17:32:01.000Z"],
@"decimal": @"12345.6789"}));
CBL_TestModel* model2 = [[CBL_TestModel alloc] initWithNewDocumentInDatabase: db];
model2ID = model2.document.documentID;
CBL_TestModel* model3 = [[CBL_TestModel alloc] initWithNewDocumentInDatabase: db];
model3ID = model3.document.documentID;
model.other = model3;
model.others = @[model2, model3];
// Verify the property getters:
CAssertEq(model.number, 1337);
CAssertEqual(model.str, @"LEET");
CAssertEqual(model.strings, (@[@"fee", @"fie", @"foe", @"fum"]));
CAssertEqual(model.data, [@"ASCII" dataUsingEncoding: NSUTF8StringEncoding]); | -1.552781 | -0.97081 | -0.724039 | -0.259752 | [
-1.5582601367401039,
-1.5473027333081137
] | [
-0.9224625332233918,
-1.0191564766141166
] | [
-0.7066022522198153,
-0.741475566318849
] | [
-0.1439907452028487,
-0.3755126990773554
] | 1,024 | [
512,
512
] | [
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] | 1,332,988 | 3,072 | Github | -1 | -1 |
CAssertEqual(model.date, [NSDate dateWithTimeIntervalSinceReferenceDate: 392773252]);
CAssertEqual(model.dates, dates);
CAssertEqual(model.decimal, decimal);
CAssertEq(model.other, model3);
CAssertEqual(model.others, (@[model2, model3]));
// Save it and make sure the save didn't trigger a reload:
CAssertEqual(db.unsavedModels, @[model]);
NSError* error;
CAssert([db saveAllModels: &error]);
CAssertEq(model.reloadCount, 0u);
// Verify that the document got updated correctly:
NSMutableDictionary* props = [model.document.properties mutableCopy];
CAssertEqual(props, (@{@"number": @(1337),
@"str": @"LEET",
@"strings": @[@"fee", @"fie", @"foe", @"fum"],
@"data": @"QVNDSUk=",
@"date": @"2013-06-12T23:40:52.000Z",
@"dates": @[@"2013-06-12T23:40:52.000Z", @"2013-06-13T17:32:01.000Z"],
@"decimal": @"12345.6789",
@"other": model3.document.documentID,
@"others": @[model2.document.documentID, model3.document.documentID],
@"_id": props[@"_id"],
@"_rev": props[@"_rev"]}));
// Update the document directly and make sure the model updates:
props[@"number"] = @4321;
CAssert([model.document putProperties: props error: &error]);
CAssertEq(model.reloadCount, 1u);
CAssertEq(model.number, 4321);
// Store the same properties in a different model's document:
[props removeObjectForKey: @"_id"];
[props removeObjectForKey: @"_rev"];
CAssert([model2.document putProperties: props error: &error]);
// ...and verify its properties:
CAssertEq(model2.reloadCount, 1u);
CAssertEq(model2.number, 4321);
CAssertEqual(model2.str, @"LEET");
CAssertEqual(model2.strings, (@[@"fee", @"fie", @"foe", @"fum"]));
CAssertEqual(model2.data, [@"ASCII" dataUsingEncoding: NSUTF8StringEncoding]);
CAssertEqual(model2.date, [NSDate dateWithTimeIntervalSinceReferenceDate: 392773252]);
CAssertEqual(model2.dates, dates);
CAssertEqual(model2.decimal, decimal);
CAssertEq(model2.other, model3);
CAssertEqual(model2.others, (@[model2, model3]));
CAssertEqual(model2.others, model.others);
CAssertEqual($cast(CBLModelArray, model2.others).docIDs, (@[model2.document.documentID,
model3.document.documentID]));
}
{
// Close/reopen the database and verify again:
db = reopenTestDB(db);
CBLDocument* doc = [db documentWithID: modelID];
CBL_TestModel* modelAgain = [CBL_TestModel modelForDocument: doc];
CAssertEq(modelAgain.number, 4321);
CAssertEqual(modelAgain.str, @"LEET");
CAssertEqual(modelAgain.strings, (@[@"fee", @"fie", @"foe", @"fum"]));
CAssertEqual(modelAgain.data, [@"ASCII" dataUsingEncoding: NSUTF8StringEncoding]);
CAssertEqual(modelAgain.date, [NSDate dateWith | -1.139139 | -0.545983 | -0.135314 | 0.235809 | [
-1.1637936131884608,
-1.1144852977445054
] | [
-0.6341713316325271,
-0.4577944163735198
] | [
-0.10234282792201324,
-0.16828509458200425
] | [
0.13955861403671566,
0.33206001388742906
] | 1,024 | [
512,
512
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] | 1,332,988 | 4,096 | Github | -1 | -1 |
TimeIntervalSinceReferenceDate: 392773252]);
CAssertEqual(modelAgain.dates, dates);
CAssertEqual(modelAgain.decimal, decimal);
CBL_TestModel *other = modelAgain.other;
CAssertEqual(modelAgain.other.document.documentID, model3ID);
NSArray* others = modelAgain.others;
CAssertEq(others.count, 2u);
CAssertEq(others[1], other);
CAssertEqual(((CBL_TestModel*)others[0]).document.documentID, model2ID);
}
[db _close];
}
TestCase(API_SaveMutatedSubModel) {
NSError* error;
CBLDatabase* db = createEmptyDB();
CBL_TestModel* model = [[CBL_TestModel alloc] initWithNewDocumentInDatabase: db];
CBL_TTestMutableSubModel* submodel = [[CBL_TTestMutableSubModel alloc] initWithFirstName: @"Jens" lastName: @"Alfke"];
model.mutableSubModel = submodel;
NSMutableDictionary* props = [model.propertiesToSave mutableCopy];
[props removeObjectForKey: @"_id"];
CAssertEqual(props, (@{@"mutableSubModel":@{@"first": @"Jens", @"last": @"Alfke"}}));
[model save: &error];
CAssertNil(error);
submodel.firstName = @"Pasin";
submodel.lastName = @"Suri";
props = [[model propertiesToSave] mutableCopy];
[props removeObjectForKey: @"_id"];
[props removeObjectForKey: @"_rev"];
CAssertEqual(props, (@{@"mutableSubModel":@{@"first": @"Pasin", @"last": @"Suri"}}));
[model save: &error];
CAssertNil(error);
submodel = model.mutableSubModel;
submodel.firstName = @"Wayne";
submodel.lastName = @"Carter";
props = [[model propertiesToSave] mutableCopy];
[props removeObjectForKey: @"_id"];
[props removeObjectForKey: @"_rev"];
CAssertEqual(props, (@{@"mutableSubModel":@{@"first": @"Wayne", @"last": @"Carter"}}));
[model save: &error];
CAssertNil(error);
}
TestCase(API_ModelAttachments) {
// Attempting to reproduce https://github.com/couchbase/couchbase-lite-ios/issues/63
CBLDatabase* db = createEmptyDB();
NSError* error;
NSData* attData = [@"Ceci n'est pas une pipe." dataUsingEncoding: NSUTF8StringEncoding];
CBLDocument* doc;
{
CBL_TestModel* model = [[CBL_TestModel alloc] initWithNewDocumentInDatabase: db];
doc = model.document;
model.number = 1337;
CAssert([model save: &error], @"Initial failed: %@", error);
[model setAttachmentNamed: @"Caption.txt" withContentType: @"text/plain" content: attData];
CAssert([model save: &error], @"Save after adding attachment failed: %@", error);
// Ensure that document's attachment metadata doesn't have the "follows" property set [#63]
NSDictionary* meta = model.document[@"_attachments"][@"Caption.txt"];
CAssertEqual(meta[@"content_type"], @"text/plain");
CAssertEqual(meta[@"length"], @24);
CAssertNil(meta[@"follows"]);
model.number = 23;
CAssert([model save: &error], @"Save after updating number failed: %@", error);
}
{
CBL_TestModel* model = [CBL_TestModel modelForDocument: doc];
CAssertEq(model.number, 23);
CBLAttachment* attachment = [model attachmentNamed: @"Caption.txt"];
CAssertEqual(attachment.content | -1.377463 | -0.661031 | -0.276155 | 0.082328 | [
-1.6240045573320443,
-1.1309214028924905
] | [
-0.9332063047112501,
-0.38885521599309564
] | [
-0.5742421737075738,
0.021932982321815946
] | [
-0.1465921154711016,
0.311249051741406
] | 1,024 | [
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] | 1,332,988 | 5,120 | Github | -1 | -1 |
SARAH SYMMES, daughter of Capt. William Symmes of Charlestown; born 1652; married 7 November 1672 Rev. Moses Fiske, pastor of the church in Braintree, then including the present town of Quincy.
Moses Fiske was born in Wenham, 1643; graduated Harvard College 1662 in the class with the renowned Solomon Stoddard, of Northampton; was ordained at Braintree, now Quincy, 11 September 1672, being the third minister of that place; and was pastor there thirty-six years, till his death 20 August 1708, aged 66.
"This excellent person was ordained pastor of the church in Braintree in September 1672, in which sacred employment he continued till his dying day, a diligent, faithful laborer in the harvest of Jesus Christ; studious in the Holy Scriptures, having an extraordinary gift in prayer above many good men; and in preaching equal to the most, inferior to few; zealously diligent for God and the good of men; one who thought no labor, cost or suffering too dear a price for the good of his people. His public preaching was attended with convincing light and clearness, and powerful, affectionate application; and his private oversight was performed with humility and unwearied diligence. He lived till he was near sixty-five years of age, beloved and honored of the most that knew him. On the 18th of July, being the Lord's day, he preached all day in public, but was not well. The distemper continued and proved a malignant fever. Small hope of his recovery being entertained, his church assembled together and earnestly besought the Great Shepherd of the sheep, that they might not be deprived of him. But Heaven had otherwise determined, for on Tuesday, August 10 [equivalent to August 21, New System], he died about one in the afternoon and was with suitable solemnity and great lamentation interred in Braintree, in his own tomb, the 12th day."
The town of Braintree voted June 18, 1672, to give Mr. Fiske, by a town tax, the sum of sixty pounds in money as a yearly salary, with the use of a house to be kept in good repair by the town, and six acres of land to be fenced by them. In 1674 his salary for that year was increased to eighty pounds. During his ministry one hundred and forty-seven members were added to the church. The baptisms were seven hundred and seventy-nine.
Mrs. Sarah (Symmes) Fiske, his first wife, died 2 December 1692 having borne him fourteen children. His second wife, to whom he was married January 7, 1700-1, was Ann (Shepard) Quincy, born 1663, daughter of Rev. Thomas Shepard who has been already mentioned as a colleague of Rev. Zechariah Symmes, of Charlestown. She was the widow of Daniel Quincy, born 1681, son of the second Edmund Quincy, of Braintree. She died July 24, 1708, aged 45, less that three weeks before his own decease.
WILLIAM SYMMES, eldest son of Captain William and Mary Symmes, of Charlestown; born 1678; married Ruth Convers 7 December 1704. She was born in Woburn 28 May 1686 and was the eldest daughter of Captain Josiah and Ruth (Marshall) Convers, of that place. Though living in two separate towns, the two families were near neighbors.
Mr. Symmes was a clothier by trade, as we learn from some old papers. He had the whole of his father's large landed property. Some of it came by inheritance, and some by purchase from the other heirs. Until 1754 it was regarded as being in Charlstown; but in that year it was annexed to Medford and is now in Winchester.
"The northerly tract is bounded on the south by the north line of Medford and the southerly bounds of Mr. Symmes's farm, west by the line that divides Mr. Symmes's from | 0.759231 | 0.825191 | 0.122115 | 0.018595 | [
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Mr. Gardiner's farm, north by the line of Woburn and Stoneham, east by Malden line".
The reasons assigned were the contracted limits of Medford, containing only about two thousand acres, surrounded almost wholly by Charlestown, and the fact that the inhabitants of the northerly tract [ the Symmes family, &c. ] were but two miles from the Medford meeting-house where they attended meeting without paying for the privilege; while they were obliged to go seven miles to attend town meetings, training &c., in Charlestown.
Tradition reports that the land included in the Symmes farm was formerly the abode of a portion of the tribe of Indians called by the euphonious name of Aberginians. It is said that it contained twenty-seven wigwams. The story is likely to be true, for here were Mystic Pond and the Aberjona River [here known as "Symmes's River"], both very convenient for fishing. Nanepashemit, the sachem of the larger tribe called the Pawtuckets, whose sway extended to the Merrimack River, and who was killed in 1619 in an attack upon his tribe by the Tarratines from the Penobscot River, lived in the near vicinity, somewhere on Mystic or Aberjona River. It was his son, Sagamore John, of Mystic [Medford], who, before his death at Medford on 5 December 1633, wished to go to the God of the Christian people. The widow of Nanepashemit, in 1639, sold to the town of Charlestown all the land on the west of Mystic Pond, bounded north by Increase Nowell's lot (the Gardiner farm), west by Cambridge Common, south by the land of Mr. Cooke. This seems to have included the Symmes farm, for after her death it was claimed on 25 March 1662 by William Symmes, son of Rev. Zechariah. Or rather, the claim was for the land at the upper end of the Pond, which the squaw-sachem had reserved for her use and the use of the Indians, to plant and hunt upon, "and the weare above the Pond for the Indians to use in fishing", during her life. This "weare" must have been in the Symmes farm. [Brook's History of Medford, p. 72 et seq.] It is where the Aberjona River enters the Pond, and we are sure that this river at that place and for some distance north divided the Symmes farm from the lot of Increase Nowell.
Mr.Symmes built a clothing mill on the Aberjona River, near where the railroad bridge now crosses that stream. It was a little north of the spot where, not long ago, Mr. Robert Bacon's dam stood. A little island in the small pond near the railroad bridge shows where the waste-way was. His house was on the left bank, or eastern side of the river, nearly opposite the house of Mr. John Bacon, son of Robert Bacon. He afterwards built a large house on the spot where John Bacon's house now stands. This was on the west side of the river; it was occupied by his sons Timothy and John after him. His grandson John Symmes was born there. The first house built on the farm, where Captain William Symmes probably once lived, was further north on the west side of the river and very near the old line between Charlestown and Woburn.
William Symmes of Charlestown, gentleman, was surety on 24 March 1726 with John Richardson of Medford, for Elizabeth Richardson, widow of Captain James Richardson, late of Woburn.
His papers, still in existence, show him to have been a man of business and of influence. His farm had been reduced to eighty acres at the time of his death. This was caused by his having conveyed portions of it to his sons during his life-time, the deeds not having effect till after his death.
He died May 24, 1764, aged 86. His wife Ruth died March 16, 1758. The gravestones of both are standing in the old cemetery in Woburn.
His will is dated 27 November 1761; proved 16 April 1766; recorded Midd. Prob. Records, xxix. 192. He calls himself William Symmes, of | 0.306553 | 1.233454 | -0.120097 | 0.133055 | [
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] | 1,332,990 | 1,024 | C4 | 0 | 3 |
Medford, yeoman. He leaves legacies to his sons Zechariah, Josiah, Timothy, John and William, and his daughter Mary Munroe. To his sons Zechariah, Josiah, Timothy and John he gives his dwelling=house, barn, the mill, and about eighty acres of land; the land to be equally divided among these four sons. The portion of each is particularly described and cannot conveniently be noted here. To his son William he gives the whole expense he, the father, had incurred for his education at school and at college, and �13 6s 8d besides -- equivalent to forty dollars.
Inventory of his estate -- Real, �490 3s 4d; Personal, �33 16s 1d.
TIMOTHY SYMMES, brother of the preceding and second son of Captain William Symmes, of Charlestown; born about 1683; married Elizabeth (Collamore) Rose, July 31, 1710, widow of Jeremiah Rose and daughter of Captain Anthony Collamore, of Scituate.
Weymouth, which was the home of his mother and his home for many years, is but a few miles from Scituate. We are not surprised, therefore, at finding him there in 1707, nor at the fact that he spent the remainder of his life there.
I have before me a letter from him to his brother William, dated Scituate 28 June 1707. He sympathizes with him on the loss they had lately sustained (the death of their step-father, Rev. Samuel Torrey of Weymouth, who died 21 April 1707) and proceeds: "My heart's desire and prayer to God is that he would make us sensible of our sins against him, which provoke him to .... remove him who was so eminently serviceable for Christ and his kingdom. We all have great cause to say, 'Against Thee, Thee only have we sinned', &c. Let us fly to Christ for mercy and pardon. He has promised that he will hearken to our cries and pardon our iniquities, though great". He then reverts to his temporal affairs, speaks of working at a trade, and of his master as exceedingly kind, and loth to part with him, but as not wishing to hinder him in any plans he may make for his own advantage. "For reasonable terms", he says, "I shall depart". He then proposes that his brother meet him on the ensuing Wednesday, to talk over his plans for the future. He thinks of going to Woburn to settle in three or four weeks. At the close he says: "Give my duty unto uncle and aunt, and my kind salutations to the lady of my best affections, Miss R.B."
The uncle and aunt probably were his mother's brother and sister. The "Miss R.B." he did not marry, as it seems.
He at length settled on a farm near the centre of South Scituate, Mass., on the Boston road, where his grandson John Cleves Symmes visited him in 1762. He died in 1765, aged 82.
ZECHARIAH SYMMES, brother of the preceding and third son of Captain William Symmes of Charlestown; born 168... Unmarried.
About all we know of him is derived from a letter written by him to his brother William Symmes and wife, dated 21 January 1706 which is Old Style and is equivalent to 1 February 1707, New Style. From this letter it appears that about three weeks previously he had sailed from Boston in a vessel commanded by Captain Mears, with a cargo of farm produce, such as onions, cranberries, &c., suited to a West India market. The vessel could not have been of large size, since he mentions as officers only Captain Mears and a mate, a brother of the captain. Some days after sailing, a conspiracy was discovered to take possession of the vessel after first taking the lives of Captain Mears, his brother, and young Symmes. There were, he says, three blood-thirsty men who had this design, two of them Frenchmen and the third a runaway, a deserter from the navy. The design having been discovered, Captain Mears, his brother and young Symmes armed themselves, took possession of all the ammunition, drove the conspirators below, | 0.623633 | 0.56555 | 0.094216 | 0.08493 | [
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and kept them prisoners under the hatches eight or nine days, until they came under the guns of a fort in Jamaica, when they delivered them to a British man-of-war, receiving better men in their room. He ascribes his deliverance to the mercy of God. The letter breathes the language of ardent affection for his brothers and sisters, and for his "honoured parents", who at this time were Mr. and Mrs. Torrey of Weymouth. He says nothing about a wife, and it is probable he had none.
NATHANIEL SYMMES, brother of the preceding and youngest son of Captain William Symmes; born about 1690. His mother, in a petition to the Probate Court in March 1692-3, prays that she may be appointed guardian to her youngest child, Nathaniel, in order that she may have legal power to improve the mill stream given to said child, by making lease of the same till said child becomes of age. The mill privilege seems, however, never to have come into his immediate possession. His brother William bought it.
He became of age about 1710 or 1711. We infer this from some receipts before us of money paid by William Symmes in November 1712 to Israel Walker and Oliver Noyes, who had supplied Nathaniel with goods out of their stores in Boston.
He was a cordwainer in Boston and was living in 1720 when his mother, Mrs. Mary Torrey, in her will made him her executor and residuary legatee. We have no further information respecting him. It is not known whether he had a family.
ZECHARIAH SYMMES, eldest son of Rev. Zechariah Symmes of Bradford, and grandson of Rev. Zechariah Symmes of Charlestown; born in Bradford 13 March 1674; married Dorcas Brackenbury 28 November 1700.
He was of Charlestown and died between 1709 and 1713. His widow Dorcas signed a deed on 4 March 1712-13. [Brief details of their son Zechariah appear under "Fourth Generation". Their other children were Dorcas, John Brackenbury, and William].
It is altogether probable that there are, or have been, descendants of this family. I have not found them, except they be found in the following schedule, which I make out from the Malden records, as copied in the New England Genealogical Register, vol. xi. pp. 129, 130, 211, 213.
I am fully persuaded that 'John Simms' of Malden, the father of these children, is identical with John Brackenbury Symmes, second son of Zechariah Symmes and Dorcas Brackenbury.
Rev. THOMAS SYMMES, brother of the preceding and second son of Zechariah Symmes of Bradford; born there 1 February 1677-8; married first Elizabeth Blowers of Cambridge, a sister of the Rev. Thomas Blowers of Beverly. She died 6 April 1714. He married second, Hannah Pike 28 March 1715, daughter of Rev. John Pike of Dover, N.H. She died in childbed 1 February 1718-19. He married third, Eleanor (Thompson) Moody, 19 January 1720-1. She was born 9 November 1679 and was daughter of Benjamin Thompson, and granddaughter of Rev. William Thompson, first minister of Braintree, and widow of Eleazar Moody of Dedham. She survived her second husband.
He was instructed in the rudiments of the Latin language by his father. His preparation for college was completed at Charlestown under the able tuition of Mr. Emerson, a distinguished teacher. He was admitted to Harvard College in 1694 and graduated there in 1698, decidedly the first scholar in his class. He remained two years longer at Cambridge to perfect his education, through aid received from Mr. Brattle and other benefactors. He was ordained 30 December 1702, the first minister of Boxford. During his pastorate there of only six years, seventy-two persons were added to that church. He met with difficulties, however | 0.38394 | 0.775053 | 0.114506 | 0.285235 | [
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, the nature of which is not known; but they greatly tried his patience and led him in 1706 to think of a removal.
By the death of his father, March 1707-8, the way was opened for his resettlement in Bradford, a town joining Boxford, and he was installed there December 1708. His salary was one hundred pounds, paid in a depreciated currency. The smallness of his salary subjected him to great embarrassments, so that he could not bring up any of his sons to college as he wished, though they possessed promising talents.
He was a minister at Bradford nearly seventeen years and during that time two hundred and thirty persons were added to that church. At one time, 11 June 1723, two hundred and thirty-four persons united in the communion. In the year 1720, fifty-nine persons were admitted, forty-six of them in three months and twenty-five in one day. At the time of his death there were but one hundred and twenty families in the town.
He was a man of earnest piety. His walk was close with God, as appears from documents now in existence. He was very conscientious and punctual in the duty of secret prayer - considering this, with the daily reading of the Bible, an eminent means of security from temptations. In all his difficulties and straits he had recourse to a prayer-hearing God. He had faith in the covenant of grace. In one place he says: "I found much comfort and encouragement in pleading the covenant with God, urging the prayers laid up for me in heaven, offered by my godly ancestors. My dear children! if you ever see this, remember that you are children of many prayers. But trust not to that: pray for yourselves". May his posterity remember this! With prayer he united fasting, observing sometimes stated and sometimes occasional seasons for seeking the divine direction and blessing.
In sacred music he took great delight and was himself a good singer. To this exercise he attended in his own family, on the morning and evening of every Sabbath; and in the latter part of his life, every day. He did what he could to reform the practice of singing in public worship, which had been very low. He introduced many new tunes and preached on the subject. He published, in 1722, a "Joco-Serious Dialogue concerning Regular Singing". It is full of wit and sarcasm and was designed to ridicule the opposers of what he calls "regular singing", that is, singing by rule or "singing by note" which he strongly advocated in preference to the old method of "lining out the hymns" and singing by impulse. It is a tract of sixty or more pages, and he informs us that he wrote it in a single day, adding a few quotations afterward.
* The fight took place in the present town of Fryeburg, Maine, 8 May 1725 Old Style, answering to 19 May New Style. Captain John Lovewell, with thirty-three men, encountered a much superior force of Indians under the noted Paugus. The Indians took them in front and rear. The action lasted from ten A.M. till sunset, or about ten hours. Notwithstanding the great disparity of force the Indians had the worst of it and retired from the field soon after sunset. Of our thirty-three men only twelve lived to return home. Captain Lovewell and twelve of his men lay dead on the field. --[Symmes's Memoir; Sewall's History of Woburn].
"A Legacy of Advice to the Church of Bradford"
"A Monitor for Delaying Sinners"
"An Artillery Election Sermon, 1720"
"A Sermon at the Ordination of Rev. Joseph Emerson at Malden, 1721"
"A Funeral Sermon for Rev. Thomas Barnard, 1718"
He was a man of much intellectual ability, diligently cultivated by close study. His library contained many of the books of his father and grandfather, and for those days was somewhat large. He usually reviewed his classical studies once a year. In his family he sometimes fluently rendered the Hebrew Bible into English.
In religious sentiment he was thoroughly Calvinistic. He was diligent in visiting his people, especially the sick - always aiming to give the conversation a religious direction. He loved to preach and embraced every opportunity for performing this service | 1.370106 | 1.120645 | 1.069401 | 0.524561 | [
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A csehek Magyarországban Jósika Miklós történelmi regénye, az Abafi mellett a legsikeresebbnek tartott műve. Először Pesten adták ki 1839-ben, négy kötetben.
Cselekménye
Hunyadi Mátyást a magyarok királyává választják. Küldöttség hozza haza Prágából, vele együtt érkezik Budára gyermekkori játszótársa, Izabella. A küldöttség tagja Zokoli Mihály lovag is, aki útja közben megismeri és megszereti Giskra leányát, Serenát.
Giskra cseh vezér felvidéki rablói, Komoróczi Péter és társai Magyarországon garázdálkodnak. A fiatal Mátyás király rendet akar teremteni és Rozgonyi Sebestyént küldi ellenük; ebben a seregben szolgál Zokoli Mihály, a rablók kérlelhetetlen üldözője. Ő dúlta föl Komoróczi rablólovag fészkét, aki bosszúból koholt levelekkel árulás gyanújába keveri. Zokoli Mátyás hű támasza ugyan, de gyanú vetült rá, ezért távozik a királyi udvarból. Elhatározza, hogy addig nem emeli fel sisakrostélyát, amíg ártatlanságában akár egyetlen lovag is kételkedik.
Ettől kezdve Elemér, a sas néven folytatja harcát a felvidéki haramiák ellen. Giskrát Mátyás király hűségére téríti, végül egy lovagi tornán az ártatlanságát egyedül kétségbevonó, már földönfutóvá lett Komoróczit is legyőzi. Zokoli, a nevét és becsületét visszaszerzett lovag elnyeri Giskra leányának, Serenának kezét, és a rablóvilágnak vége. Maga Mátyás király feleségének halála után titokban szintén házasságra lép kedvesével, a szép Izabellával.
Jellemzése
A mese három szálon bonyolódik és nem kristályosodik ki egyetlen főhős körül, ahogy például az Abafiban. A regény középpontjában Mátyás király alakja áll, őt látjuk uralkodásának első hét évében, szerelmesei karjában és a Giskrával viselt háború körülményei között. A másik két cselekményszál főhősének személye költött. Az olvasó érdeklődése elsősorban Zokoli Mihályra irányul, az ő vitézi tettei, harca becsülete helyreállításáért, és Serena iránt érzett szerelme a mese legfontosabb szála. A harmadik szál a rablólovag Komoróczi története, aki sziklavárából garázdálkodik és Zokoli életére tör. A két személy sorsa összekapcsolódik, de a történelmi eseményekhez és Mátyás uralkodói tetteihez csak gyönge szálakkal kapcsolódik.
Sok valószínűtlen részlete, laza kompozíciója és a középkori lovagvilág sablonos rajza ellenére ez Jósika Miklós egyik legsikerültebb munkája. Itt is sok a romantikus elem (rablók fészke, álöltözet, nőszöktetés, titkos várfenék stb.); a jellemrajz hiányosabb, de a korrajz jóval szélesebb, mint | 0.03433 | -0.942159 | -0.91394 | -0.249346 | [
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History's Strongest Senior Brother
HSSB420: Great awkwardness
"His movement speed is very high, but there doesn't seem to be any powerful aura-qi fluctuations," Ah Hu listened, then said uncomprehendingly, "Could it be some special sort of movement technique? How strange."
Xu Fei said, "That's very possible. The martial legacies here must be dissimilar to those within our Eight Extremities World somewhat."
Yan Zhaoge listened to the movements of the newcomer as well, "It's not surprising. After all, they're two different worlds. Rather than the difference in martial arts, I'm actually more concerned about how different the language here is from that in our world."
"After all, even in our Eight Extremities World, there are many different dialects used in different places that differ greatly. If not for a standardised, common one that we all use as a whole, it would be very hard for those originating from different places to understand what the other side is saying."
Yan Zhaoge smacked his lips, "Wanting to find someone through whom to understand the situation of this world, if we cannot communicate, that would be awkward indeed."
Ah Hu asked, "Young Master, why don't you conceal yourselves first. I'll stay and interact with this person."
"The other party's strength is currently completely unknown to us, being very hard to determine as well. He might even possess some strange secret arts that we do not know of."
Having always followed by Yan Zhaoge's side, other than being his follower, Ah Hu had also assumed the role of his bodyguard for a long time.
While Yan Zhaoge's cultivation base was currently higher than his, Ah Hu's mindset was still to keep Yan Zhaoge's safety as the first and utmost priority.
While he usually appeared lazy, when he got serious, this big guy was extremely reliable in carrying out his duties.
Being in a completely new and unknown world, everything was foreign, maybe even dangerous.
Currently, Ah Hu's heart as a follower of Yan Zhaoge was already strung up completely tightly.
Yan Zhaoge waved his hands, "If even so many of us are unable to cope with this danger, you alone going would be no different from suicide."
"Relax a bit. While we are currently unable to determine the cultivation base of this newcomer, it shouldn't be too high as well."
"One art connects, all arts connect. In the great overarching dao of this world, all arts converge as one at their base. While there are differences amongst them, the higher up one goes, the more similarities there are to be found," Yan Zhaoge gazed into the distance, "It's just like us and the Flame Devils. The feeling when a Martial Saint or a Flame Devil King is standing in front of you is completely different from the feeling given off by a Martial Grandmaster or a Great Flame Devil."
"This is unless the other party conceals it intentionally or you instead identify an Essence Talisman Martial Grandmaster as an Essence Spirit Martial Grandmaster or vice versa, but it is absolutely always possible to distinguish whether it is a Martial Saint or a Martial Grandmaster."
Yan Zhaoge and the others temporarily concealed themselves within the dense forest. Very quickly, a figure flashed, appearing before them.
It was a tall middle-aged man, the colour of his skin like bronze, his eyes circular with a head like a leopard's and rough features.
His body seemed to contain a flourishing, explosive power, his eyesight appearing exceptionally acute as light shot out in all directions.
Yan Zhaoge nodded to himself, "At least he has a human appearance, not being much different from us like some mutant species."
As he observed the big man, he saw the man's gaze suddenly flicker as he then promptly looked over in the direction of Ying Longtu.
The people dispersed within the dense forest all frowned.
While Ying Longtu's cultivation base was the lowest of them all, he possessed the Big Dipper physique, able to freely control his qi and blood as he liked as he also had a mind that was clear and without flaw. If he intentionally hid himself, it should be very difficult for others to detect his presence.
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roked his lower chin, "Is it that this person, deeply concealed, is stronger than I thought, or is it that due to some special reason, his sensory abilities are extremely high?"
After scanning the location at which Ying Longtu was concealed, the big man now turned to look at where Feng Yunsheng was.
Yan Zhaoge coughed lightly, showing himself.
Seeing Yan Zhaoge, that man's face was filled with caution and wariness. No longer looking in the directions of Feng Yunsheng and Ying Longtu, he instead stared fixatedly at Yan Zhaoge.
Yan Zhaoge spread his hands apart, first asking, "How do I address you?"
He first used the standardised common language of the Eight Extremities World.
This was not a shot in the dark. Instead, the common language of the Eight Extremities World had propagated from words left behind from before the time of the Great Calamity, most forming a system, also being the closest to the civilisation of before the Great Calamity.
In terms of intonation, he also tried to go as close to that of olden times as he could.
While if it were Yan Zhaoge appraising it, there was still quite the difference, it was not through luck or coincidence that this language and its written characters were mainstream in the current Eight Extremities World.
If this world before him was also descended from the civilisation of before the Great Calamity, they should share some commonalities.
In the end, having heard him speak, the big man showed no reaction. Instead, he appeared uncomprehending as he came to look even more wary.
It was clearly the look of not having understood anything at all.
Yan Zhaoge frowned, considering for a moment before asking once more, "How do I address you?"
This time, Feng Yunsheng, Ah Hu and the others all found his words foreign, seeming weird and hard to grasp in some places as they were not easy to understand.
This time, Yan Zhaoge had directly gone on to use the common language of before the Great Calamity.
To make an analogy, as compared to the language of the current Eight Extremities World, this was like the difference between modern language and that of olden times. Some words had similar meanings, but if one had not learnt them before in a systemic manner, it would be very difficult to understand their complete meaning.
In other words, it was like that of all the characters being individually recognisable yet not making any sense when stringed together.
By Yan Zhaoge's conjecture, the martial arts his mother cultivated in were a comparatively complete martial legacy of before the Great Calamity. Therefore, the language and written characters employed back then had probably not been lost as well.
This time, the big man seemed to react a little, looking somewhat baffled at the same time that he remained wary.
Yan Zhaoge felt rather helpless under that gaze, which seemed to be asking:
Can this guy just speak properly?
Observing carefully, Yan Zhaoge felt that the reaction of this person when hearing the pre-Great Calamity spoken language resembled that of Xu Fei, Feng Yunsheng and the others.
"In that case, it is most probable that the language and written characters here, like that of the Eight Extremities World, also originate from the pre-Great Calamity civilisation, but have undergone their own unique development as they are now completely different from that currently employed in the Eight Extremities World."
Yan Zhaoge gradually came to an understanding.
However, despite this, he felt a bit of a headache coming on.
With them being unable to successfully communicate through language, great awkwardness was felt by him.
"Anyway, since it's definitely developed from the pre-Great Calamity language, as he says a few words and I make a few guesses, I might be able to understand the general gist of it," Yan Zhaoge sighed, "If it really isn't possible, I suppose I'd have to employ something like hand signals, even though that would be rather embarrassing."
Thinking such, Yan Zhaoge said a few more words using the olden language of before the Great Calamity.
That big man looked uncomprehendingly and warily at him once more.
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Guest Lecture- Mr. Aniruddha Khekale, Group Director-HR, Emerson
On November 17, 2016 November 17, 2016 By iimkashipurblogIn UncategorizedLeave a comment
Indian Institute of Management Kashipur was graced by the presence of Mr. Aniruddha Khekale, Group Director- HR, Emerson. Mr. Khekale visited IIM Kashipur to deliver a guest lecture on HR in VUCA times.
"People have taken VUCA as a reality. What began in 2008, has been continuing" said Mr. Khekale.
Mr. Khekale discussed the challenges of the HR role in the modern times when dynamics is the key. He reiterated on the essence of restructuring and reorganizing the organization from time to time to tackle problems more effectively and efficiently.
"HR should be the moral compass of an organization. Many CEOs consider themselves to be the HR leader of their respective organizations" said Mr. Khekale.
Mr. Khekale concluded the speech by sharing his own experience in the role of an HR leader and answering questions from the students.
The entire student fraternity of IIM Kashipur thanks MR. Anirudda Khekale for providing us with such an insightful session.
Guest Lecture By Mr. Shekhar Chandra Kumar, Ex-COO, Air India Express
On November 17, 2016 By iimkashipurblogIn UncategorizedLeave a comment
Indian Institute of Management Kashipur was graced by the presence of Mr. Shekhar Chandra Kumar, Ex-COO, Air India Express. Mr. Shekhar visited IIM Kashipur to deliver a guest lecture on the opportunities in the aviation industry.
Mr. Shekhar started the lecture by introducing the aviation industry to the students of IIM Kashipur from his perspective, the challenges and the vicissitude of opportunities that lay forth.
He partitioned his lecture into three phases, discussing the pre-liberalization phase, the liberalization phase and the post liberalization phase of the aviation industry. The session was an extremely informative one and opened up new avenues of opportunity that now lies before the students of IIM Kashipur.
He also mentioned a case of excellence, a project on air travel demand in the Kumaon region of Uttarakhand that was conducted by the 2014 graduates of IIM Kashipur under the guidance of Mr. Shekhar Chandra Kumar, that found its way to the table of Air India, and using which demand was speculated for Pantnagar airport.
The Operations enthusiastic fraternity of IIM Kashipur, which is nestled in one of the most densely industrialized areas in South-East Asia, made the session and extremely engaging one with students questioning about every nook and corner of opportunity for operational excellance in the aviation industry in general and Air India in particular.
The entire student fraternity of IIM Kashipur thanks Mr. Shekhar Chandra Kumar for providing us with valuable time to share such information and insight.
Kacofonia 2016
On November 3, 2016 By iimkashipurblogIn UncategorizedLeave a comment
Indian Institute of Management Kashipur organized the inter-sectional wars of PGP 2016-2018 batch students on 29th of October, 2016. The coveted title of being called the sectional wars champion of Kacofonia 2016 spurred all the students of IIM Kashipur to conjure their talents and bring the best of their performance to the battle field.
After taking out preparation times from the hectic time schedule of their IIM Kashipur lives, students hustled forth to the auditorium every night to hone their battle axes and wielding skills.
The battle of the sections had already begun on Facebook, YouTube and other social media sites.
The stage was set and the warriors were ready to bring the fury upon their opponents.
Kacofonia 2016 was spread over four different talent bundles, namely, Literary, Music, Dance, Theatre and Categorically Insane.
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Catherine M. Beattie (Mrs. Harold) Representative of Danville, Democrat, was born at Danville on April 5, 1921. Occupation: Farming. She was educated at Danville elementary and High School. She is married and has three sons and seven daughters. Member of the following organizations: Order of Eastern Star; Danville Woman's Club; Pythian Sisters; Danville Parent-Teachers Group; Caledonia County Farm Bureau. Member of the House of 1965.
Robert T. Gannett of Brattleboro, Windham County. Republican, was born in Boston, Massachusetts on September 26, 1917 and became a resident of present town in August, 1946. He was married to the former Sarah Alden Derby who died in 1999, and they have one daughter and two sons. Occupation: lawyer. He was educated in elementary schools in Milton, Massachusetts; Milton Academy and Harvard University (B.A., 1939; L.L.B., 1942). Member of Brattleboro Retreat, former trustee; Brattleboro Memorial Hospital, corporator and past president; National Life Insurance Company, director emeritus, United Way of Windham County, former director, Vermont Community Foundation, trustee; American College of Probate Counsel.
He served in the U.S. Army, 1942-46; Major, Field Artillery. Member of the House: 1953, 1955. 1957, 1959. Member of the Senate: 1973-74, 1975-76, 1977-78, 1979-80, 1981-82, 1983-84, 1985-86, 1987-88, 1989-90, 1991-92.
John G. Kristensen of Guilford, Windham County, Democrat, was born in New York, New York on March 31, 1924 and became a resident of present town in 1952. Occupation: attorney. He was educated in elementary schools in Valley Stream, New York; Central High School, Valley Stream, New York; Ursinus College, Collegeville, Pennsylvania (B.S., 1974); Harvard Law School, Cambridge, Massachusetts (LL.B., 1950). Member of: Judicial Conduct Board, 1987-93; Town Moderator, 1975-present; Brattleboro Union High School Moderator, 1984-present; Guilford Town Democratic Committee; Vermont Bar Association, president, 1985-86; Vermont State Board of Education, 1961-67; Guilford School Board, 1956-69; former member of state and county Democratic committees. He served as a law clerk for federal judge and former Governor Ernest W. Gibson, 1952-53. He is married to the former Calista R. Elliot, and they have three daughters and two sons. Member of the House: 1993-94 (seated February 2, 1993), 1995-96, 1997-98.
Madeleine M. Kunin of Shelburne, Chittenden County, Democrat, was born in Zurich, Switzerland on September 28, 1933 and became resident of present town in September, 1957. Occupation: teacher/writer. She was educated in elementary school in Forest Hills, New York; Pittsfield, Massachusetts High School; University of Massachusetts (Amherst), (B.A., 1956); School of Journalism, Columbia University (New York), M.S., 1957; University of Vermont, M.A., 1967; Fellow, Institute of Politics, Harvard University, 1983; Honorary degrees: Norwich University, St. Michael's College, Middlebury College, University of Massachusetts (Amherst), Green Mountain College, and Marlboro College. 1987 Chair, New England Governor's Conference; Executive Committee member, National Governor's Association; Executive Committee, National Democratic Governor's Association. Member of the House from District 1- | -0.527408 | 0.180117 | -0.50941 | -0.833354 | [
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Precision and brawn a deadly combination
A bizarre, pouched super-predator that terrorised South America millions of years ago had huge sabre-like teeth but its bite was weaker than that of a domestic cat, new research shows.
The huge canine teeth of Thylacosmilus atrox extended almost into its braincase
Australian and American marsupials are among the closest living relatives of the extinct Thylacosmilus atrox, which had tooth roots extending rearwards almost into its small braincase.
"Thylacosmilus looked and behaved like nothing alive today," says UNSW palaeontologist, Dr Stephen Wroe, leader of the research team.
"To achieve a kill the animal must have secured and immobilised large prey using its extremely powerful forearms, before inserting the sabre-teeth into the windpipe or major arteries of the neck – a mix of brute force and delicate precision."
The iconic North American sabre-toothed 'tiger', Smilodon fatalis, is often regarded as the archetypal mammalian super-predator.
However, Smilodon - a true cat - was just the end point in one of at least five independent 'experiments' in sabre-tooth evolution through the Age of Mammals, which spanned some 65 million years.
Thylacosmilus atrox is the best preserved species of one of these evolutionary lines - pouched sabre-tooths that terrorised South America until around 3.5 million years ago.
For its size, its huge canine teeth were larger than those of any other known sabre-tooth.
Smilodon's killing behaviour has long attracted controversy, but scientists now mostly agree that powerful neck muscles, as well as jaw muscles, played an important role in driving the sabre-teeth into the necks of large prey.
Little was known about the predatory behaviour in the pouched Thylacosmilus.
To shed light on this super-predator mystery, Dr Wroe's team of Australian and US scientists constructed and compared sophisticated computer models of Smilodon and Thylacosmilus, as well as a living conical-toothed cat, the leopard.
These models were digitally 'crash-tested' in simulations of biting and killing behaviour. The results are published in the journal PLoS ONE.
"We found that both sabre-tooth species were similar in possessing weak jaw-muscle-driven bites compared to the leopard, but the mechanical performance of the sabre-tooths skulls showed that they were both well-adapted to resist forces generated by very powerful neck muscles," says Dr Wroe.
"But compared to the placental Smilodon, Thylacosmilus was even more extreme."
"Frankly, the jaw muscles of Thylacosmilus were embarrassing. With its jaws wide open this 80-100 kg 'super-predator' had a bite less powerful than a domestic cat. On the other hand - its skull easily outperformed that of the placental Smilodon in response to strong forces from hypothetical neck muscles."
"Bottom line is that the huge sabres of Thylacosmilus were driven home by the neck muscles alone and - because the sabre-teeth were actually quite fragile - this must have been achieved with surprising precision."
"For Thylacosmilus - and other sabre-tooths - it was all about a quick kill."
"Big prey are dangerous – even to super-predators – and the faster the kill the less likely it is that the predator will get hurt – or for that matter attract unwanted attention from other predators."
"It may not have been the smartest of mammalian super-predators – but in terms of specialisation – Thylacosmilus took the already extreme sabre-tooth lifestyle to a whole new level," says Dr Wroe.
Media contacs: Dr Stephen Wroe: +61 2 4969 3006 [email protected]; UNSW Science media: Deborah Smith: +61 2 9385 7307 [email protected]
Climate change, not human activity, led to megafauna ext | 0.67705 | 2.601495 | 1.023748 | 1.008416 | [
0.8140177062857492,
0.5400826204859972
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3.00975783133275,
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1.1962125770747325,
0.851283797622472
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1.0656464295347425,
0.9511861377316155
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] | 1,333,017 | 0 | CommonCrawl | 7 | 9 |
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