text:
\section{Introduction}

Since the study of harmonic maps was initiated in 1964 by Eells and Sampson \cite{ES}, the theory of harmonic maps has been widely investigated and developed. It is well-known that harmonic maps between two Riemannian manifolds $(M, g)$ and $(N, h)$ are critical points of the energy functional
$$E(\varphi) = \frac{1}{2} \int_M |d \varphi |^2 dv_g,$$
where $\varphi \in C^\infty (M, N)$ and $dv_g$ denotes the volume element of $g$. The Euler-Lagrange equation of the energy functional $E$ is given by
$$\tau (\varphi) = {\rm tr}_g \nabla d \varphi = 0,$$
where $\tau (\varphi)$ is called the {\it tension field} of $\varphi$. In 1983, as a natural extension of harmonic maps, Eells and Lamaire \cite{EL} proposed to study the $k$-harmonic maps which are critical points of the functional
$$E_k (\varphi) = \frac{1}{2} \int_M \| (d+d^*)^k \varphi \|^2 dv_g$$
for any $k\in \mathbb{N}$. In particular, when $k=2$, the functional $E_2$ is given by
$$E_2 (\varphi)= \frac{1}{2} \int_M |\tau (\varphi)|^2 dv_g,$$
which is called the {\it bienergy} functional. Jiang \cite{Jiang} derived the first and the second variational formulas of the bienergy functional. The Euler-Lagrange equation of $E_2$ is given by
$$\tau_2 (\varphi)={\rm tr}_g (\nabla^\varphi \nabla^\varphi - \nabla^\varphi_{\nabla^M}) \tau(\varphi) - {\rm tr}_g R^N (d\varphi, \tau(\varphi)) d\varphi = 0,$$
where $\nabla^\varphi$ denotes the pull-back connection on $\varphi^{-1}TN$ and $R^N$ denotes the curvature operator on $(N, h)$. Here $\tau_2 (\varphi)$ is called the {\it bitension} field of $\varphi$.
A smooth map $\varphi: (M, g) \rightarrow (N, h)$ is called a {\it biharmonic} map if $\tau_2(\varphi)=0$. We note that the above biharmonic map equation is a system of the fourth-order semilinear elliptic partial differential equations.

If an isometric immersion $\varphi: (M, g) \rightarrow (N, h)$ is biharmonic, then $M$ is called a {\it biharmonic  submanifold} in $N$. In the recent decades, there has been an increasing attention on biharmonic submanifolds and a lot of important results on biharmonic submanifolds have been obtained. We refer the readers to two well-written books \cite{Ou-Chen} by Ou-Chen and \cite{Urakawa} by Urakawa for recent developments on this area (see also a recent survey article \cite{Ou2015} by Ou and references therein). In his pioneering work of submanifolds of finite type in a Euclidean space, B.-Y. Chen \cite{Chen1984, Chen1991} provided an alternative definition of biharmonic submanifolds of a Euclidean space by using harmonic mean curvature vector field. According to his definition, $M$ is called a biharmonic submanifold if an isometric immersion $\varphi: M \rightarrow \mathbb{R}^n$ satisfies
$$\Delta_M {\bf H} = 0,$$
where $\Delta_M$ denotes the Laplacian of $M$ and ${\bf H}$ denotes the mean curvature vector field on $M$. It is obvious that these two definitions are equivalent in case where the ambient space is a Euclidean space. It is clear that a harmonic map is automatically a biharmonic map and hence a minimal submanifold is a biharmonic submanifold. For this reason, a non-harmonic map is called a {\it proper} biharmonic map and similarly a non-minimal biharmonic submanifold is called a {\it proper} biharmonic submanifold. There are some known examples of proper biharmonic  submanifolds in the unit sphere (see \cite{BMO2009, CMO, CMO2002, Jiang, Ou2006, Sasahara} for instance). See also \cite{Ou2010, Ou2012} for examples of proper biharmonic hypersurfaces in conformally flat spaces.

In contrast, it is unknown whether there exists a proper biharmonic submanifold in a Euclidean space. In this point of view, Chen \cite{Chen1991} in 1991 proposed the following well-known conjecture.
\begin{conj}[Chen's conjecture]  \label{conj: Chen}
Any biharmonic submanifold in a Euclidean space is minimal.
\end{conj}

\noindent Chen \cite{Chen1991} in 1985 and Jiang \cite{Jiang1987} in 1987 independently proved that any biharmonic surface in $\mathbb{R}^3$ is minimal. In 1995, Hasanis-Vlachos \cite{HV} proved that any biharmonic hypersurface in $\mathbb{R}^4$ is minimal. Defever \cite{Defever} in 1998 gave a different proof of the result by Hasanis and Vlachos. Very recently, Fu-Hong-Zhan \cite{FHZ} showed that Chen's conjecture is true for biharmonic hypersurfaces in $\mathbb{R}^5$ by transferring the biharmonic equations into a system of algebraic differential equations. See also \cite{AM, AGR, BMO2008, BMO, BMO2013, CMO, CMO2002, CM, FLO, Fu2014, Fu2015a, Fu2015b, LMO, Luo2014, Luo2015, Maeta2014, Maeta2017, MO, NU2011, NU2013, Ou2010, Ou2012, OT2012, Wheeler} for partial results on Chen's conjecture under additional geometric conditions on biharmonic submanifolds.

The main purpose of this paper is to prove Chen's conjecture. In Section 3, we study some rigidity properties of bihamronic hypersurfaces in an Einstein manifold. From the invariant equations for biharmonic hypersurfaces due to Ou \cite{Ou2010}, we derive a fundamental identity which the mean curvature function on biharmonic hypersurfaces in an Einstein manifold satisfies (see Proposition \ref{lem2021-5-15-4}). Using this identity, we are able to prove Chen's conjecture for biharmonic hypersurfaces in an Einstein manifold with nonpositive scalar curvature as follows:

\vspace{.12in}
\noindent
{\bf Theorem A} (see Theorem \ref{thm2021-5-15-10})
Any biharmonic hypersurface in an Einstein manifold with nonpositive scalar curvature is minimal.

\vspace{.12in}
Moreover we prove that any compact biharmonic hypersurface (without boundary) in an Einstein manifold must have constant mean curvature (see Theorem \ref{thm: compact hypersurface in Einstein}). In Section 4, we study biharmonic submanifolds in space forms.  We derive a fundamental identity on the mean curvature vector field on biharmonic submanifolds in space forms (see Proposition \ref{prop: fundamental for submanifolds}) from the invariant equations for biharmonic submanifolds due to  Balmu\c{s}-Montaldo-Oniciuc \cite{BMO2013}. By using this identity, we show that any biharmonic submanifold in a space form of nonpositive sectional curvature is minimal (see Theorem \ref{thm : biharmonic submanifolds in space forms}). As a consequence, we give a complete solution to Chen's conjecture as follows.

\vspace{.12in}
\noindent
{\bf Theorem B} (see Corollary \ref{cor: Chen})
Any biharmonic submanifold in a Euclidean space is minimal.

\vspace{.12in}

On the other hand, Chen's conjecture can be generalized to more general ambient spaces. The following conjecture was proposed by Caddeo-Montaldo-Oniciuc \cite{CMO} in 2001.
\begin{conj}[the generalized Chen's conjecture] \label{conj: generalized Chen}
Any biharmonic submanifold in a Riemannian manifold with nonpositive sectional curvature is minimal.
\end{conj}

There are many affirmative partial results on the generalized Chen's conjecture. For example, Caddeo-Montaldo-Oniciuc \cite{CMO2002} showed that any biharmonic submanifold in $3$-dimensional hyperbolic space $\mathbb{H}^3$ is minimal. Later, Balmu\c{s}-Montaldo-Oniciuc \cite{BMO} proved that the generalized Chen's conjecture is true for biharmonic hypersurfaces in $\mathbb{H}^4$. See also \cite{BMO2008, Luo2015, NU2011, NU2013} for related results about the generalized Chen's conjecture with additional geometric conditions. Despite a lot of affirmative partial answers to Conjecture \ref{conj: generalized Chen}, it turned out that this conjecture is false by Ou-Tang \cite{OT2012} in 2012. Indeed, they constructed a family of counterexamples of $4$-dimensional proper biharmonic hypersurfaces in a conformally flat space with negative sectional curvature. It should be mentioned that these counterexamples are incomplete biharmonic submanifolds in the ambient space with nonconstant nonpositive sectional curvature. In Section 4, we give an affirmative partial answer to the generalized Chen's conjecture as follows:

\vspace{.12in}
\noindent
{\bf Theorem C} (see Corollary \ref{cor: Chen for H^n})
Any biharmonic submanifold in a hyperbolic space ${\Bbb H}^n$ is minimal.

\vspace{.12in}

On the contrary, although examples of proper biharmonic submanifolds in spheres are rather rich as mentioned before, they all have constant mean curvature. Motivated by this observation, Balmu\c{s}-Montaldo-Oniciuc \cite{BMO2008} proposed the following.
\begin{conj}[BMO conjecture] \label{conj: BMO}
Any biharmonic submanifold in $\mathbb{S}^n$ has constant mean curvature.
\end{conj}

\noindent Recently, Maeta-Ou \cite{MO} proved that a complete hypersurface $M^n$ with constant scalar curvature in a sphere is biharmonic if and only if it is minimal or it has nonzero constant mean curvature and $|A|^2=n$, where $|A|^2$ denotes the squared norm of the second fundamental form. In \cite{FLO}, Fetcu-Loubeau-Oniciuc showed that Conjecture \ref{conj: BMO} is true for compact proper biharmonic hypersurfaces in a sphere with constant scalar curvature. See also \cite{BMO2008, BMO, CMO, CMO2002, CMP, Maeta2017} for more results on biharmonic submanifolds in a sphere. Finally we give a partial answer to BMO conjecture for compact biharmonic submanifolds in a sphere.

\vspace{.12in}
\noindent
{\bf Theorem D} (see Corollary \ref{cor: BMO})
Any compact biharmonic submanifold  in a sphere has constant mean curvature.

\section{Preliminaries}

We begin with some basic facts and notions about biharmonic submanifolds. Let $(M^m, g)$ and $(N^n, h)$ be Riemannian manifolds of dimension $m$ and $n$, respectively. Let $\varphi: M\rightarrow N$ be an isometric immersion. For each $p\in M$, the tangent space $T_{\varphi (p)} N$ splits as an orthogonal direct sum
$$T_{\varphi (p)} N = d\varphi (T_p M) \oplus d\varphi (T_p M)^\perp$$
and the normal bundle of $M$ in $N$ is given by $NM=\bigcup_{p\in M} d\varphi (T_p M)^\perp$. We denote by $\nabla$ and $\nabla^N$ the Levi-Civita connections on $M$ and $N$, respectively, and by $\nabla^\varphi$ the induced connection on the pull-back bundle $\varphi^{-1}TN= \bigcup_{p\in M} T_{\varphi (p)} N$. Recall that the bienergy of $\varphi$ is defined by
$$E_2 (\varphi)= \frac{1}{2} \int_M |\tau (\varphi)|^2 dv_g,$$
where $\tau(\varphi)={\rm tr}_g \nabla d\varphi$ is the tension field of $\varphi$ which vanishes for a harmonic map. The Euler-Lagrange equation associated to $E_2$ is given by
$$\tau_2 (\varphi) = \Delta^\varphi \tau (\varphi)- {\rm tr}_g R^N (d\varphi, \tau(\varphi))d\varphi=0,$$
where $\tau_2 (\varphi)$ is called the {\it bitension field} of $\varphi$ and $R^N$ is the Riemannian curvature tensor of $(N, h)$ given by
$$R^N (X, Y)Z=\nabla^N_X \nabla^N_Y Z - \nabla^N_Y \nabla^N_X Z - \nabla^N_{[X, Y]} Z$$
for any tangent vector fields $X, Y, Z$ of $N$. Here we use the following sign convention:
$$\Delta^\varphi \tau (\varphi) = {\rm tr}_g (\nabla^\varphi \nabla^\varphi - \nabla^\varphi_{\nabla^M}) \tau (\varphi).$$
As $\varphi: M \rightarrow N$ is an isometric immersion, we are able to identify $d\varphi (X)$ with $X\in C(TM)$ at any point $p\in M$ and identify $M$ with its image, where $C(TM)$ denotes the space of the sections of the tangent bundle $TM$. We also denote by $\langle \cdot, \cdot \rangle$ the induced metric $\varphi^{-1}h$. The shape operator $A_\nu$ for a unit normal vector field $\nu$ on $M$ is defined by
$$A_\nu X = -(\nabla^N_X \nu)^\top$$
for all $X\in C(TM)$ and $\nu \in C(NM)$. With these notations in mind, we write the Gauss and Weingarten formulas as
\begin{align*}
\nabla^N_X Y &= \nabla_X Y + B(X, Y) \\
\nabla^N_X \nu &=-A_\nu X + \nabla^\perp_X \nu,
\end{align*}
where $B$ denotes the second fundamental form and $\nabla^\perp$ denotes the normal connection on the normal bundle $NM$ of $M$ in $N$. Moreover, we have
$$\langle B(X, Y), \nu\rangle = \langle A_\nu X, Y\rangle$$
for all $X, Y \in C(TM)$ and $\nu\in C(NM)$. For any $p\in M$, we choose a local field of orthonormal frames $\{e_1, \cdots, e_m, e_{m+1}, \cdots, e_n\}$ in $N$ such that, restricted to $M$, the vectors $e_1, \cdots, e_m$ are tangent to $M$ (and consequently, the remaining vectors $e_{m+1}, \cdots, e_n$ are normal to $M$). Then the second fundamental form can be written as
$$B(X, Y) = \sum^n_{\alpha=m+1} B_\alpha (X, Y) e_\alpha.$$
The (normalized) {\it mean curvature vector field} ${\bf H}$ of $M$ at $p$ is given by
$${\bf H}(p) = \frac{1}{m}{\rm tr}_g B = \frac{1}{m} \sum^m_{i=1} B(e_i, e_i) = \sum^n_{\alpha=m+1} H_\alpha (p)e_\alpha,$$
where the function $H_\alpha$ is defined by
$$H_\alpha (p)=\frac{1}{m} \sum^m_{i=1} B_\alpha (e_i, e_i)= \frac{1}{m} \sum^m_{i=1} \langle A_{e_\alpha} (e_i), e_i\rangle.$$
We note that the tension field $\tau(\varphi)$ of $\varphi$ can be written as
\begin{align} \label{eqn: tension field}
\tau(\varphi)=m {\bf H}.
\end{align}
Therefore we see that the submanifold $M$ in $N$ is biharmonic if and only if
\begin{align} \label{eqn: biharmonic eqn}
\Delta^\varphi {\bf H} - \sum^m_{i=1} R^N (d\varphi (e_i), {\bf H}) d\varphi(e_i)=0.
\end{align}

\section{Biharmonic hypersurfaces in an Einstein manifold}

In this section, we study rigidity properties of biharmonic hypersurfaces in an Einstein manifold. For this purpose, let $\varphi : M^m \to (N^{m+1}, h)$ be an isometric immersion with the unit normal vector field $\nu$. Choose a local orthonormal frame $\{e_1, \cdots, e_m\}$ on $M$. By (\ref{eqn: tension field}), the tension field $\tau(\varphi)$ is given by
$$
\tau(\varphi) = \sum_{i=1}^m \left(\overline{\nabla}_{e_i} e_i\right)^\perp = m H\nu,
$$
where $H$ denotes the mean curvature and $\overline{\nabla}$ denotes the Levi-Civita connection on $N$. The following result due to Y.-L. Ou \cite{Ou2010} is the invariant form of the biharmonic equations (\ref{eqn: biharmonic eqn}) for biharmonic hypersurfaces in a Riemannian manifold.

\begin{lem}[Ou \cite{Ou2010}]\label{thm2021-5-15-11}
Let $\varphi:  M^m \to N^{m+1}$ be an isometric immersion of codimension one with mean curvature vector ${\bf H}=H\nu$. Then $\varphi$ is biharmonic if and only if it satisfies
\begin{eqnarray}
\left\{\begin{array}{ll}
\Delta H - H|A|^2 + H \,{\rm Ric}^N(\nu, \nu) = 0,\\
2A_\nu(\nabla H) + m H \nabla H - 2H \left({\rm Ric}^N(\nu)\right)^\top =0,
\end{array}\right. \label{eqn2021-5-15-1}
\end{eqnarray}
where $\Delta$ and $\nabla$ denote the Laplacian and the gradient on $M$, respectively. Here ${\rm Ric}^N: T_q N \rightarrow T_q N$ denotes the Ricci operator of the ambient space $N$ defined by $\langle {\rm Ric}^N(X), Y\rangle = {\rm Ric}^N(X, Y)$, and $\nu$ is the unit normal vector field on $M$.
\end{lem}

In fact, both equations in (\ref{eqn2021-5-15-1}) can be derived by decomposing (\ref{eqn: biharmonic eqn}) into the tangential and normal parts. The divergence of the second equation in (\ref{eqn2021-5-15-1}) gives a fundamental identity for the mean curvature function on biharmonic hypersurfaces in Einstein manifolds, which is quite simple but turns out to be very useful.
\begin{prop}\label{lem2021-5-15-4}
Let $\varphi: M^m \to N^{m+1}$ be a biharmonic isometric immersion of a hypersurface $M$ with mean curvature vector ${\bf H}=H\nu$ into an Einstein manifold $N$. Then
\begin{eqnarray}
3|\nabla H|^2 + H \Delta H =0.\label{eqn2021-5-15-2}
\end{eqnarray}
\end{prop}

\begin{proof}
Observe that  ${\rm Ric}^N(\nu)^\top = 0$, since $N$ is Einstein. Taking the divergence of the second equation in (\ref{eqn2021-5-15-1}), we have
$$
2\,{\rm div} A_\nu (\nabla H) + m |\nabla H|^2 + m H \Delta H = 0.
$$
Now fix a point $p\in M$ and choose a local orthonormal frame $\{e_1, e_2, \cdots, e_m, \nu\}$ of $N$
 which is normal at $p$ so that $\left(\overline{\nabla}_{e_\alpha}e_\beta\right)|_p = 0$, where $\alpha, \beta = 1, 2, \cdots, n+1$ and $e_{m+1} = \nu$. Here $\overline{\nabla}$ denotes the Levi-Civita connection on $N$. Recall that $A_\nu (X) = - (\overline{\nabla}_X \nu)^\top$ for all $X \in C(TM)$. Then
it follows from definition together with Ricci identity that, at $p\in M$,
\begin{eqnarray*}
{\rm div}A_{\nu}(\nabla H) &=& \langle \nabla_{e_i}A_\nu(\nabla H), e_i\rangle\\
&=& - \langle \nabla_{e_i}\overline{\nabla}_{\nabla H}\nu, e_i\rangle\\
&=&
-e_i\langle \overline{\nabla}_{\nabla H}\nu, e_i\rangle + \langle \overline{\nabla}_{\nabla H}\nu, \overline{\nabla}_{e_i}e_i\rangle\\
&=&
e_i \langle \overline{\nabla}_{\nabla H}e_i, \nu\rangle - \langle \overline{\nabla}_{\nabla H} \nabla_{e_i}e_i, \nu\rangle\\
&=&
\langle \overline{\nabla}_{e_i}\overline{\nabla}_{\nabla H}e_i, \nu\rangle + \langle \overline{\nabla}_{\nabla H}e_i, \overline{\nabla}_{e_i} \nu\rangle -  \langle \overline{\nabla}_{\nabla H} \nabla_{e_i}e_i, \nu\rangle\\
&=&
\langle \overline{\nabla}_{\nabla H}\overline{\nabla}_{e_i}e_i, \nu\rangle
+ \langle \overline{\nabla}_{[e_i, \nabla H]} e_i, \nu\rangle + \langle R^N(e_i, \nabla H)e_i, \nu\rangle\\
&&\quad
+ \langle \overline{\nabla}_{\nabla H}e_i, \overline{\nabla}_{e_i} \nu\rangle -  \langle \overline{\nabla}_{\nabla H} \nabla_{e_i}e_i, \nu\rangle\\
&=&
\langle \overline{\nabla}_{\nabla H} \left(\overline{\nabla}_{e_i}e_i\right)^\perp, \nu\rangle
+ \langle \overline{\nabla}_{[e_i, \nabla H]} e_i, \nu\rangle - {\rm Ric}^N(\nabla H, \nu)\\
&&\quad
+ \langle \overline{\nabla}_{\nabla H}e_i, \overline{\nabla}_{e_i} \nu\rangle -  \langle \overline{\nabla}_{\nabla H} \nabla_{e_i}e_i, \nu\rangle,
\end{eqnarray*}
where we used the Einstein summation convention. Thus, at the point $p \in M$, we see that
\begin{eqnarray*}
{\rm div}A_{\nu}(\nabla H)
&=&
\langle \overline{\nabla}_{\nabla H} \left(\overline{\nabla}_{e_i}e_i\right)^\perp, \nu\rangle\\
&=&
\langle \overline{\nabla}_{\nabla H} \left( m H\nu \right), \nu\rangle \\
&=& m |\nabla H|^2.
\end{eqnarray*}
Since the point $p$ was chosen arbitrarily, we obtain
$$
3|\nabla H|^2 + H \Delta H = 0.
$$
\end{proof}

Applying Proposition~\ref{lem2021-5-15-4}, we are able to prove Chen's conjecture for biharmonic hypersurfaces in an Einstein manifold with nonpositive scalar curvature.

\begin{thm}\label{thm2021-5-15-10}
Any biharmonic hypersurface in an Einstein manifold with nonpositive scalar curvature is minimal.
\end{thm}

\begin{proof}
Let $M^m$ be an $m$-dimensional biharmonic hypersurface in an Einstein manifold $(N^{m+1}, h)$ with ${\rm Ric}^N = \lambda h$ for some constant $\lambda \leq 0$, where ${\rm Ric}^N$ denotes the Ricci curvature tensor of $N$. Note that the constant $\lambda$ is given by $\lambda=\frac{{\rm Scal}^N}{m+1}$, where ${\rm Scal}^N$ denotes the scalar curvature of $N$.

The first equation in (\ref{eqn2021-5-15-1}) becomes
\begin{align}
\Delta H - H |A|^2 +\lambda H = 0. \label{eqn: 1}
\end{align}
Combining (\ref{eqn: 1}) with Proposition \ref{lem2021-5-15-4}, we obtain
$$
0\leq 3|\nabla H|^2 + H^2 |A|^2 = \lambda H^2 \leq 0,
$$
which implies that $H=0$. This completes the proof of Theorem \ref{thm2021-5-15-10}.
\end{proof}

As another application of Proposition \ref{lem2021-5-15-4}, we prove that any compact biharmonic  hypersurface (without boundary) in an Einstein manifold has constant mean curvature.
\begin{thm} \label{thm: compact hypersurface in Einstein}
Any compact biharmonic hypersurface $M^m$ in an Einstein manifold $N^{m+1}$ has constant mean curvature. In particular, if the Einstein manifold $N$ has nonpositive scalar curvature, then $M$ is minimal.
\end{thm}

\begin{proof}
Since $\Delta H^2 = 2H\Delta H + 2 |\nabla H|^2$, it follows from Proposition \ref{lem2021-5-15-4} that
$$
\Delta H^2 = -4 |\nabla H|^2.
$$
This implies that the function $H^2$ is superharmonic and hence it is constant because $M$ is compact. Since $H$ is constant,  the first equation in (\ref{eqn2021-5-15-1}) shows that
\begin{align}
H\left(|A|^2 - {\rm Ric}^N(\nu,\nu)\right) = 0. \label{eqn: 2}
\end{align}
In particular, when the Einstein manifold $N$ has nonpositive scalar curvature, we may assume that $H$ is a nonzero constant. Then, from (\ref{eqn: 2}), it follows that
$$
0 \le |A|^2 = {\rm Ric}^N(\nu, \nu) \le 0,
$$
which $\nu$ is the unit normal vector field on $M$. This shows that $A= 0$ and thus $H=0$, which is a contradiction.
\end{proof}

\noindent As a corollary, we give an affirmative partial answer to Conjecture \ref{conj: BMO} (BMO conjecture) as follows.

\begin{cor}
Any compact biharmonic hypersurface in a sphere has constant mean curvature.
\end{cor}

In general, when the target manifold is not necessarily Einstein, we obtain the following result, which is a generalized version of Proposition \ref{lem2021-5-15-4}.

\begin{prop}\label{lem2021-5-15-9}
Let $\varphi: M^m \to N^{m+1}$ be a biharmonic isometric immersion with mean curvature vector ${\bf H}=H\nu$ into a Riemannian manifold  $N$. Then
\begin{align*}
3m |\nabla H|^2 + m H \Delta H - 2{\rm Ric}^N(\nabla H, \nu) - H \langle \nabla {\rm Scal}^N, \nu\rangle +
2H \overline{\nabla}_\nu {\rm Ric}^N (\nu, \nu) =0,
\end{align*}
where ${\rm Scal}^N$ denotes the scalar curvature of $N$.
\end{prop}

\begin{proof}
Taking the divergence of the second equation in (\ref{eqn2021-5-15-1}) gives
\begin{eqnarray}
2\,{\rm div}A_\nu (\nabla H) + m |\nabla H|^2  + m H \Delta H -2 \,{\rm Ric}^N(\nabla H, \nu) - 2\,H {\rm div}{\rm Ric}^N(\nu)^\top = 0.\label{eqn2021-5-15-8}
\end{eqnarray}
As in the proof of Proposition~\ref{lem2021-5-15-4}, we have
\begin{align}
{\rm div} A_\nu(\nabla H) = m |\nabla H|^2. \label{eqn: a}
\end{align}
Now fix a point $p\in M$ and choose a local frame $\{e_1, e_2, \cdots, e_m, \nu\}$ of $N$
 which is normal at $p$ so that $(\overline{\nabla}_{e_\alpha}e_\beta)|_p = 0$, where $\alpha, \beta = 1, 2, \cdots, m+1$ and $e_{m+1} = \nu$.
 Since ${\rm Ric}^N(\nu)^\top = {\rm Ric}^N(\nu, e_i)e_i$,  we compute, at the point $p$,
\begin{eqnarray}
{\rm div}{\rm Ric}^N(\nu)^\top &=& \langle \nabla_{e_k} {\rm Ric}^N(\nu)^\top, e_k\rangle \nonumber \\
&=&
\left\langle\nabla_{e_k}\left(\langle R^N(\nu, e_i)e_i, e_j \rangle e_j\right), e_k\right\rangle \nonumber \\
&=&
e_k \left\langle \langle R^N(\nu, e_i)e_i, e_j\rangle, e_k\right\rangle - \langle R^N(\nu, e_i)e_i, e_j\rangle
\langle e_j, \nabla_{e_k}e_k\rangle \nonumber \\
&=&
e_k\langle R^N(\nu, e_i)e_i, e_k\rangle  - {\rm Ric}^N(\nu, e_j)\langle e_j, \nabla_{e_k}e_k\rangle\nonumber \\
&=&
e_k \left({\rm Ric}^N(\nu, e_k)\right) = \nabla_{e_k} {\rm Ric}^N (\nu, e_k) \nonumber\\
&=&
{\rm div}{\rm Ric}^N (\nu) - \overline{\nabla}_{\nu} {\rm Ric}^N(\nu, \nu) \nonumber \\
&=&
\frac{1}{2}\langle \overline{\nabla}{{\rm Scal}^N}, \nu\rangle - \overline{\nabla}_{\nu} {\rm Ric}^N(\nu, \nu), \label{eqn: b}
\end{eqnarray}
where we used the identity $2\,{\rm div} {\rm Ric}^N = d \, {\rm Scal}^N$ in the last equality. Substituting (\ref{eqn: a}) and (\ref{eqn: b}) into (\ref{eqn2021-5-15-8}), we get the conclusion.
\end{proof}

Applying Proposition \ref{lem2021-5-15-9}, we prove the following rigidity result for biharmonic hypersurfaces in a Riemannian manifold with nonpositive parallel Ricci curvature and with constant scalar curvature, which can be regarded as a generalization of Theorem \ref{thm2021-5-15-10}.

\begin{thm}
Let $M^m$ be a biharmonic hypersurface in a Riemannian manifold $N^{m+1}$. Assume that $N$ has nonpositive parallel Ricci tensor and constant scalar curvature. Then $M$ is minimal.
\end{thm}

\begin{proof}
Since $\overline{\nabla} {\rm Scal}^N = 0$ and $\overline{\nabla}_\nu {\rm Ric}^N = 0$ by assumption, combining the first equation of (\ref{eqn2021-5-15-1}) with Proposition \ref{lem2021-5-15-9} yields that
$$
3|\nabla H|^2 + H^2 |A|^2 = 2\,{\rm Ric}^N(\nabla H, \nu) + H^2 {\rm Ric}^N(\nu, \nu).
$$
Since $N$ has nonpositive Ricci curvature, we conclude that $H=0$.
\end{proof}

\section{Biharmonic submanifolds in space forms}

In this section, we consider  biharmonic submanifolds of higher codimension in space forms. We start with the invariant equations for biharmonic submanifolds in a Riemannian manifold due to Balmu\c{s}-Montaldo-Oniciuc \cite{BMO2013}, which is a higher-codimensional version of Lemma \ref{thm2021-5-15-11}.

\begin{lem}[Balmu\c{s}-Montaldo-Oniciuc \cite{BMO2013}]
Let $\varphi: M^m \to N^{n}$ be an isometric immersion of an $m$-dimensional submanifold $M$ into an $n$-dimensional Riemannian manifold $N$ with the mean curvature vector field {\bf H}. Then $\varphi$ is biharmonic if and only if it satisfies
\begin{eqnarray}
\left\{\begin{array}{ll}
-\Delta^\perp {\bf H} + {\rm tr}B(\cdot, A_{\bf H} \cdot) + {\rm tr}\left(R^N(d\varphi(\cdot), {\bf H})d\varphi(\cdot)\right)^\perp= 0,\\
2\,{\rm tr}A_{\nabla^\perp_{(\cdot)} {\bf H}} + m |{\bf H}|{\rm grad}|{\bf H}| + 2\,{\rm tr} \left(
R^N(d\varphi(\cdot), {\bf H})d\varphi(\cdot)\right)^\top= 0,
\end{array}\right. \label{eqn2021-5-15-12}
\end{eqnarray}
where $A$ denotes the shape operator on $M$, $B$ denotes the second fundamental form on $M$, and $\Delta^\perp$ and $\nabla^\perp$ denote the Laplacian and connection on the normal bundle of $M$, respectively.
\end{lem}

Let $\varphi: M^m \rightarrow N^{m+n}$ be a biharmonic submanifold in a Riemannian manifold $N$, where $n \geq 2$. Choose a local orthonormal frame $\{e_1, e_2, \cdots, e_m\}$ on $M$ and $\{\nu_1, \nu_2, \cdots, \nu_n\}$ on the normal bundle $NM$ such that the mean curvature vector field ${\bf H}$ is given by
$${\bf H} = \sum_{\alpha} H_\alpha \nu_\alpha.$$
Recall that
$$
\langle B(X, Y),  {\bf H}\rangle = \langle A_{\bf H}X, Y\rangle
$$
for any tangent vectors $X, Y$ on $M$. Thus
\begin{eqnarray}
\langle B(e_i, A_{\bf H}e_i), {\bf H}\rangle
&=&
 \langle A_{\bf H}e_i, A_{\bf H}e_i\rangle \nonumber \\
&=& \langle \left(\overline{\nabla}_{e_i} {\bf H}\right)^\top, \left(\overline{\nabla}_{e_i} {\bf H}\right)^\top\rangle \nonumber  \\
&=& |\left(\overline{\nabla}_{e_i} {\bf H}\right)^\top|^2 \nonumber
\end{eqnarray}
Moreover it follows that
\begin{eqnarray*}
\langle {\bf H}, \left(R^N(e_i, {\bf H})e_i\right)^\perp\rangle
&=&
\langle {\bf H}, R^N (e_i, {\bf H})e_i \rangle \\
&=& R^N (e_i, {\bf H}, e_i, {\bf H})\\
&=&
\langle R^N(e_i, H_\alpha\nu_\alpha)e_i, H_\alpha\nu_\alpha\rangle.
\end{eqnarray*}
In case where $N$ is a space form, we have
\begin{align}
\left(R^N(e_i, {\bf H})e_i\right)^\top = \langle R^N(e_i, {\bf H})e_i, e_j\rangle e_j =0. \label{eqn: R^N vanishes}
\end{align}

In the following, we obtain a fundamental identity for the mean curvature vector field on biharmonic submanfolds in space forms, which is a key ingredient for the proof of Chen's conjecture.

\begin{prop}\label{prop: fundamental for submanifolds}
Let $\varphi: M^m \to N^{m+n}$ be a biharmonic submanifold $M$ in a space form $N$. Then the mean curvature vector field ${\bf H}$ on $M$ satisfies
$$
\langle \Delta^\perp {\bf H}, {\bf H} \rangle + 3|\nabla^\perp {\bf H}|^2 = 0.
$$
\end{prop}

\begin{proof}
By taking the divergence of the second equation in (\ref{eqn2021-5-15-12}), we obtain
$$
2\,{\rm div}\left({\rm tr}A_{\nabla^\perp_{(\cdot)} {\bf H}}\right) + m \left|\nabla|{\bf H}|\right|^2
 + m |{\bf H}|\Delta |{\bf H}| = 0,
$$
where we used the equation (\ref{eqn: R^N vanishes}) because the ambient space $N$ is a space form. Moreover, since
$$
\frac12 \Delta |{\bf H}|^2 = \langle {\bf H}, \Delta^\perp {\bf H}\rangle
+ |\nabla^\perp {\bf H}|^2,
$$
it is equivalent to
\begin{eqnarray}
2\,{\rm div}\left({\rm tr}A_{\nabla^\perp_{(\cdot)} {\bf H}}\right)
+ m \langle \Delta^\perp {\bf H}, {\bf H} \rangle + m |\nabla^\perp {\bf H}|^2 = 0.\label{eqn2021-5-16-5}
\end{eqnarray}

Fix a point $p \in M$ and choose a local orthonormal frame $\{e_1, \cdots, e_m, \nu_1, \cdots, \nu_n\}$ on $N$ which is normal at $p$, i.e., $(\overline{\nabla}_{e_\alpha}e_\beta)|_p = 0$ for any $e_\alpha$ and $e_\beta$. Here $e_\alpha = \nu_l$ for $\alpha = m+l$, where $l=1, \cdots, n$. Then, at $p\in M$,
\begin{eqnarray*}
{\rm div}\left({\rm tr}A_{\nabla^\perp_{(\cdot)} {\bf H}}\right)
&=& {\rm div}\left({\rm tr}A_{\nabla^\perp_{e_j} {\bf H}}(e_j)\right)\\
&=& \langle \nabla_{e_i} \left({\rm tr}A_{\nabla^\perp_{e_j} {\bf H}}(e_j)\right), e_i\rangle\\
&=&
- \langle \nabla_{e_i}\left(\overline{\nabla}_{e_j}(\nabla^\perp_{e_j}{\bf H})\right)^\top, e_i\rangle\\
&=&
-e_i\langle \overline{\nabla}_{e_j}(\nabla^\perp_{e_j}{\bf H}), e_i\rangle +
\langle \overline{\nabla}_{e_j}(\nabla^\perp_{e_j}{\bf H}), \nabla_{e_i}e_i\rangle\\
&=&
e_i \langle \overline{\nabla}_{e_j} e_i, \nabla^\perp_{e_j}{\bf H}\rangle -
\langle \nabla^\perp_{e_j}{\bf H}, \overline{\nabla}_{e_j}\nabla_{e_i}e_i\rangle\\
&=&
\langle \overline{\nabla}_{e_i}\overline{\nabla}_{e_j}e_i, \nabla^\perp_{e_j}{\bf H}\rangle
+ \langle \overline{\nabla}_{e_j}e_i, \overline{\nabla}_{e_i}\nabla^\perp_{e_j}{\bf H}\rangle
- \langle \nabla^\perp_{e_j}{\bf H}, \overline{\nabla}_{e_j}\nabla_{e_i}e_i\rangle\\
&=&
\langle \overline{\nabla}_{e_j}\overline{\nabla}_{e_i}e_i, \nabla^\perp_{e_j}{\bf H}\rangle
+ \langle \overline{\nabla}_{[e_i, e_j]}e_i, \nabla^\perp_{e_j}{\bf H}\rangle
+ \langle R^N(e_i, e_j)e_i, \nabla^\perp_{e_j}{\bf H}\rangle \\
&&\quad
+
\langle \overline{\nabla}_{e_j}e_i, \overline{\nabla}_{e_i}\nabla^\perp_{e_j}{\bf H}\rangle
- \langle \overline{\nabla}_{e_j}\nabla_{e_i}e_i, \nabla^\perp_{e_j}{\bf H}\rangle.
\end{eqnarray*}
Since $(\overline{\nabla}_{e_\alpha}e_\beta)|_p = 0$ and $N$ is a space form, we see that
\begin{eqnarray*}
{\rm div}\left({\rm tr}A_{\nabla^\perp_{(\cdot)} {\bf H}}\right)
&=&
\langle \overline{\nabla}_{e_j}\overline{\nabla}_{e_i}e_i, \nabla^\perp_{e_j}{\bf H}\rangle
- \langle \overline{\nabla}_{e_j}\nabla_{e_i}e_i, \nabla^\perp_{e_j}{\bf H}\rangle\\
&=& \langle \overline{\nabla}_{e_j} m {\bf H}, \nabla^\perp_{e_j}{\bf H}\rangle\\
&=& m |\nabla^\perp_{e_j}{\bf H}|^2 \\
&=& m |\nabla^\perp {\bf H}|^2.
\end{eqnarray*}
Hence it follows from (\ref{eqn2021-5-16-5}) that
$$
\langle \Delta^\perp {\bf H}, {\bf H} \rangle + 3|\nabla^\perp {\bf H}|^2 = 0,
$$
which completes the proof.
\end{proof}

Now we are ready to prove Chen's conjecture on biharmonic submanifolds in space forms of nonpositive sectional curvature.

\begin{thm} \label{thm : biharmonic submanifolds in space forms}
Any biharmonic submanifold in a space form with nonpositive sectional curvature is minimal.
\end{thm}

\begin{proof}

Let $\varphi: M^m \to N^{m+n}$ be a biharmonic isometric immersion of submanifold $M$ into a space form $N$ with nonpositive sectional curvature. Take the inner product of both sides of the first equation in  (\ref{eqn2021-5-15-12}) with the mean curvature vector field ${\bf H}$. Then we have
\begin{eqnarray*}
\langle \Delta^\perp {\bf H}, {\bf H}\rangle - |\left(\overline{\nabla}_{e_i} {\bf H}\right)^\top|^2
- H_\alpha^2 \langle R^N(e_i, \nu_\alpha)e_i, \nu_\alpha\rangle=0.\label{eqn2021-5-16-7}
\end{eqnarray*}
Combining this with Proposition \ref{prop: fundamental for submanifolds}, we obtain
\begin{eqnarray}
0 \leq |\left(\overline{\nabla}_{e_i} {\bf H}\right)^\top|^2 + 3|\nabla^\perp {\bf H}|^2 = - H_\alpha^2 \langle R^N(e_i, \nu_\alpha)e_i, \nu_\alpha\rangle \leq 0,                   \label{eqn2021-5-16-10}
\end{eqnarray}
where we used the assumption that  $N$ has nonpositive sectional curvature. Therefore it follows that ${\bf H} = 0$, which completes the proof.
\end{proof}

\noindent As a consequence, we see that Chen's conjecture (Conjecture \ref{conj: Chen}) on biharmonic submanifolds in a Euclidean space is true.

\begin{cor} \label{cor: Chen}
Any biharmonic submanifold in a Euclidean space  is minimal.
\end{cor}

\noindent Moreover even though there exist counterexamples constructed by Ou-Tang \cite{OT2012}, we are still able to give an affirmative partial answer to the generalized Chen's conjecture (Conjecture \ref{conj: generalized Chen}) in case where the ambient space is a hyperbolic space.
\begin{cor} \label{cor: Chen for H^n}
Any biharmonic submanifold in a hyperbolic space ${\Bbb H}^n$ is minimal.
\end{cor}

Finally, as another application of Proposition \ref{prop: fundamental for submanifolds}, we prove the following.

\begin{thm} \label{thm: compact submanifold}
Any compact biharmonic submanifold in a space form must have constant length of the mean curvature vector field.
\end{thm}

\begin{proof}

Let $\varphi: M^m \to N^{m+n}$ be a biharmonic isometric immersion of a compact submanifold $M$  into a space form $N$. By Proposition \ref{prop: fundamental for submanifolds}, the mean curvature vector field ${\bf H}$ satisfies
$$
\frac{1}{2}\Delta |{\bf H}|^2 + 2 |\nabla^\perp {\bf H}|^2 = 0,
$$
which shows that the function $|{\bf H}|^2$ is superharmonic on $M$. Since $M$ is compact, the maximum principle shows that $|{\bf H}|$ is constant. This completes the proof.
\end{proof}

\noindent As an immediate consequence of Theorem \ref{thm: compact submanifold}, one can show that BMO conjecture (Conjecture \ref{conj: BMO}) is true for compact biharmonic submanifolds in a sphere.

\begin{cor} \label{cor: BMO}
Any compact biharmonic submanifold  in a sphere has constant mean curvature.
\end{cor}

\vskip 0.3cm
\noindent
{\bf Acknowledgment:} The first-named author was supported by the National Research Foundation of Korea (NRF-2021R1A2C1003365) and the second-named author was supported by the National Research Foundation of Korea (NRF-2019R1A2C1004948).

redpajama_set_name: RedPajamaArXiv


Costume: Made from a Kevlar/Flameweave material it provides Good Protection against physical damage, Excellent from Heat and Fire and Typical protection from other energy forms.
Wrist Launcher: Mounted on her right arm is a wrist-launcher that fires small CO2 propelled bolts with a 3 area range. Helena carries 20 regular broad blades capable of inflicting Good damage. She can attach and fire a Grappling hook and line made from Excellent strength material.
Nightsight lenses: Excellent nightsight at 3 area range.
Pouches: Secreted in her costume Huntress has various pieces of equipment for her war on crime.
Lamborghini: This is a standard Sports car of this type, with no special modifications.
Helena Bertinelli is the daughter of Guido Bertinelli, head of the Bertinelli crime family, one of the largest Mafia families in America. She grew up in the East Town of Gotham City. At age six Helena was abducted by a person known as Vince on behalf of a rival crime family. When she was returned by the police later that evening it was quite evident that the experience had taken it's toll on the young girl. Where she was once an outgoing, optemistic, naive child she quickly became more serious, surreptitious and distrustful.
After the incident Helena's father decided that she was to be sent of to school far from the family. Though in order to secure her safety he assigned her a bodyguard much to Helena's dismay. At age 19 Helena left College for the spring break to attend her cousin Freddy's wedding leaving her bodyguard behind. Shortly after she arrived an assassin known as Omerta the Silencer broke in and murdered her entire family. Helena only survived the attack by playing dead herself.
Once it became known that she had survived a number of hitmen to were sent to stalk her. Fortunately her bodyguard, only known as Sal, found her first and brought her to safety. He then taught her to protact herself, training her in the martial arts as well as the use of all kinds of weapons. Despite she was now able to protact herself she still felt small and vulnerable, so in order to be able to bring her family's killers to justice she decided that she had to create another identity for herself. She then designed a costume and under the identity of the Huntress she began seeking out her family's killers. Eventually she found out that Omerta was actually her family's consiglieri, a lawyer who had been emplyed both by Guido Bertinelli and a Mafioso known as Mandragora. Mandragora had wanted the Bertinelli fortune for himself and had ordered the hit on the family so that he could establish a major American empire himself.
Having defeated Omerta and Mandragora, Helena decided to move to New York and keep on fighting crime as the Huntress. Following an encounter with the Justice League America she was invited to join but declined. When Maxwell Lord looked her up in her secret identity a short time later he used his mindbending abilities to make her accept the offer. She left the group again a few months later as she learned that she had been manipulated by Lord.
About two years ago she returned to Gotham City where she continued her crimefighting career. Early on she met Robin III who quickly discovered her secret identity, much to her dismay. Since their first encounter she and Robin have teamed up a number of times, and Robin has honored their freindship by not telling his mentor of her secret. Batman himself does not approve of the Huntress as he believes her methods to be to violent, and because her reckless behavior reminds him of Batgirl and her final fate. By not accepting her as one of the family he hopes to discourage her from continuing her career as Huntress, but instead it has made more determined than ever. Though Huntress has been called in to help out on rare occations such as during the Gotham Plague, she mostly keeps to her own.
Seeing that the Huntress was becoming even more violent than before in her battle against crime the Batman recently invited her to join the JLA in an attempt to soothen her aggressive side. However, after the recent destruction of Gotham City caused by the big quake has left the city in the midst of the biggest crime wave the city has ever seen Huntress has once again become more bitter and though she has not yet killed anyone she is prone to use deadly force on her enemies.


Paul and I ran around like blue-bottomed flies trying to find some vertical prayer flags in Rewalsar, which appear to be as scarce as hens teeth no matter where you are. So while I was in one shop organizing some to get made, he was down the road doing exactly the same thing! So we ended up with two sets. That's okay though – we got to watch the guy actually printing the design onto Paul's flags – a huge wooden printing block with a picture of Padmasambava and Tibetan writing on it. Interesting stuff.
weren't ignorant of the compliment she was paying us.
My new sunglasses I bought at Delhi 2 weeks ago – for the grand sum of 200 rupees – broke in half while we were sitting at a dhaba waiting for our bus to Mandi. So I went round the corner and bought some more for 60 rupees. I bet my bottom dollar these ones last for ages longer, Murphy's law being what it is.
We got to Mandi 2 hours early for our bus to Delhi, which was leaving at 7.30 pm. So we sat in a nearby dhaba, drank chai, turned down a whiskey from the owner, ignored the dirt all over the walls and floor and watched a really corny Indian television drama about Shiva and Parvati.
Naturally the bus was running on India time and therefore was late. It was actually more crowded than airplane seats – who would've thought that was possible? – and it ran out of fuel somewhere in the middle of nowhere. Fortunately, one of the crazy things about India is that there are dhaba's everywhere, and sure enough there was one a few feet down the road. So we sat there drinking chai while the poor guys in charge of the bus sucked on bucketfulls of diesel for a while.
Finally we got going again and made it to Delhi on a wing and a prayer, I think. It was a really uncomfortable ride, as our luggage wouldn't fit in the back of the bus because that was full, so we had to sit ou backpacks in the aisle and I had to sit another heavy bag on my lap. For a good part of the journey I couldn't feel my legs and every time they stopped for a meal or whatever, it took ages to get out of my seat, as the guy in front had his seat tipped right back, our luggage had to be put on Paul's seat so everybody else could get past, and I was literally trapped there for 5 minutes or so each time. When they're only stopping the bus for 15 or 20 minutes, every second counts.
Once at Delhi, we shared a mini-van with some Israelis. That would've been fine if they hadn't had huge amounts of luggage each. Two lots of enormous backpacks were on top of my foot for nearly an hour and by the time we got to Paharganj I think my body was convinced that I no longer owned a pair of feet nor a set of lower legs.
Getting to our room was bliss. Jeepers it's hot in Delhi! We staggered up 3 flights of stairs with our luggage, then threw everything down, dived under the cold shower and lay on the bed with the fan on typhoon speed. Some things in life are so worth striving for. That was one of 'em!
Up to the rooftop for breakfast, and lo and behold, there were a couple of kiwi friends of hours who themselves had just flown into Delhi. We lolled around with as much decadence as we could rustle up between us and talked for 2 or 3 hours.
Prayer flag maker, madly creating flags in a hurry for us to take home.
Our lovely Tibetan momo-making friends.
This entry was posted in Ravings - random wafflings, Ravings on India, Ravings on India 2008, Travel and tagged Delhi, India, momo, Rewalsar, Tibetan, Travel. Bookmark the permalink.


text:
Enhance Your 'Tweets' with Twitter Audio Messaging
Twitter emerged as a new-age social network (a peculiar concept in itself as social media is still considered a modern technology) in 2006 with a bold and enlightening concept of mini-blogging – 140 characters or less, to be exact.
Today, Twitter's booming popularity has inspired the development of new avenues to enhance social interaction among users, including the sharing of music files. However, Twitter's most recent advancement allows tweeters to share audio files.
So, whether your fancy is to make the punch line of a joke more vibrant, an apology more sincere or even sing happy birthday to a friend, Twitter is now the place to be seen and heard!
Check out these five different outlets, thanks to Mashable.com, to capture audio messaging on your next Twitter post.
1. Chir.ps
"After signing in with OAuth, it's just a matter of entering your name for your personalized Chir.ps page before you can get going on the recording side of things.
The on-screen recording is again, simple — just authorize Chir.ps to use your computer's microphone, hit one button to start, say your piece and then hit stop. You can preview your recording and once you're happy with it, add text and then tweet it.
While still decent, the audio quality of Chir.ps did not seem as great as other services we used, but the site does offer a listen count, so you can see how many people have hit play, which we do like."
2. AudioBoo
"Once you've signed in with OAuth and chosen your AudioBoo username, the online recording takes place in a pop out window, which also gives you the option to upload a common audio file.
You can talk (or sing, or hum, or groan or whatever it is you want to do) for five minutes, after which you can preview your ramblings. Once you're happy, give the recording a title and AudioBoo saves it down into the 'Your Boos' section of the site.
From here you can choose to share your audio to Facebook, get an embed code for it, or tweet it, which creates a short URL.
AudioBoo's sound quality is good, but there's no way to see how many people have tuned-in to listen to your broadcast, although there is the capability to comment on Boos."
Note: This technology is also available in iPhone and Android applications.
3. Twaud.io
"Recording online is easy: Just sign in with OAuth, authorize use of your microphone and you're good to go for a generous thirty minutes worth of record time, which is practically podcast length.
Just hit the record button, hit it again to stop and then press play to review whatever it is you've just uttered. Once you're done on the same screen there's a box to enter up to 100 characters of text, then just press the 'Twaudio it!' button and you're done.
Twaud.io's URLs are super short and plays automatically when someone hits the link. Those who listen to your recordings also get the option to download them, get an embed code, retweet them or comment on them, which is sent as an @reply to the creator.
Sound quality is good, and the site tells you how many listens you've racked up, which is also good for an ego boost."
4. Chirbit
"Chirbit doesn't offer OAuth sign-in as it's not designed solely for the micro-blogging service, but setting up an account is a pain-free process, all the same.
Once you're in, you can use the service to record up to two minutes of audio, as long as you've given the appropriate permission just by hitting the onscreen buttons. After you've listened back and decided you like what you hear, you can give the clip a title and then add it into a category.
To save your Chirbit, you hit send and it will then appear on your Chirbit homepage. To get sharing, hit the cog icon on your recording and the site gives you the option to post it to Facebook or Twitter, making you a nice little shortened URL in the process."
5. ShoutoMatic
"Head on over to the site, either sign up for an account, or connect via Facebook, and you're ready to start 'shouting.'
On the simple interface you just hit to record, review and save the clip. If you've signed in with Facebook it will prompt you to post it on there, but you can skip this bit.
After you've saved it, you can e-mail, download, post to Facebook or tweet your audio. If you tweet it, then ShoutoMatic makes you a neat shortened URL and adds the text 'Check this out' to your tweet, which obviously, you can choose to delete.
There is no listen counter on ShoutoMatic, but people are able to reply directly on your message page."
Tags > Social Media, twitter

redpajama_set_name: RedPajamaCommonCrawl


While you command three personnel, this personnel gains Biology and Officer, and becomes .
Vulcan Hologram created as prey by the Hirogen. Once liberated by Iden, he often set aside his logic and morality out of loyalty to his savior.


text:
SA, Tanzania to discuss bilateral, global issues
Bilateral, continental and global issues will form the basis of President Cyril Ramaphosa's state visit to Tanzania which gets underway today.
"The visit will allow the two leaders to discuss bilateral, continental and global issues, and to reaffirm their commitment to further cooperate closely on multilateral issues between the two countries.
"President Ramaphosa and his counterpart are also expected to attend and address a Business Forum arranged by the Ministries of Trade and Industry of both countries," said the Presidency following the President's arrival on Wednesday.
The state visit to the east African country, is President Ramaphosa's first since being elected President in May and is at the invitation of President John Pombe Magufuli.
President Ramaphosa and his counterpart are also expected to attend and address a Business Forum arranged by the Ministries of Trade and Industry of both countries.
President Ramaphosa will also visit the town of Morogoro, west of Dar es Salaam to express solidarity with and convey condolences to the government and people of Tanzania following a fuel tanker explosion that claimed more than 70 lives on Sunday, 11 August 2019.
This as President Ramaphosa has expressed his sincere regret at this mass loss of life and indicated South Africa's willingness to offer Tanzania any assistance that may be required following this tragedy.
President Ramaphosa said the tragedy at Morogoro is shared by South Africans for whom Morogoro holds great significance as the site of the first National Consultative Conference of the African National Congress (ANC) 50 years ago on May 1.
The Morogoro Conference brought together and advanced unity among diverse formations in the struggle against apartheid. The conference also issued a clarion call for an intensification of the armed struggle.
Morogoro is also where the educational Solomon Mahlangu Freedom College (SOMAFCO) was established by the exiled ANC in 1978.
The college provided educational and vocational skills to young South Africans who had fled the country after the 1976 Soweto uprising.
SADC Summit
Following the state visit, the President will attend the 39th Ordinary Summit of Southern African Development Community (SADC) Heads of State and Government on 17-18 August 2019.
The Summit will be held under the theme "A Conducive Business Environment for Inclusive and Sustainable Industrial Development".
The theme is in line with previous SADC Summit decisions that endorsed industrialisation as the overarching priority for the region based on the SADC Industrialisation Strategy and Roadmap (2015-2063).
The Summit will provide an opportunity for Heads of State and Government to be apprised on the overall implementation of the SADC work programme, including the Revised Regional Indicative Strategic Development Plan (2015-2020) and the SADC Industrialisation Strategy and Roadmap.
"The Summit will be updated on the status of the region's economy, health, and food security. It will provide policy direction about future strategic work of SADC and the SADC post 2020 Agenda."
At the Summit scheduled for the weekend, Chair of the SADC Organ on Politics, Defence and Security Cooperation (Zambia) will also present a report on the status of regional peace and security.
In his capacity as the SADC facilitator to the Kingdom of Lesotho, President Ramaphosa will present a report on developments in the facilitation process during the SADC Troika Summit of the Organ on Politics, Defence and Security Cooperation to be held on 16 August 2019.
The President is accompanied by International Relations and Cooperation Minister Naledi Pandor; Defence and Military Veterans Minister Nosiviwe Mapisa-Nqakula; Tourism Minister Mmamoloko Kubayi-Ngubane; Finance Minister Tito Mboweni and Mineral Resources and Energy Minister Gwede Mantashe. – SAnews.gov.za


text:
Trust Board (Directors)
Jill Donaldson Hodges
Jill has worked in education for over forty years in a variety of roles and positions. Jill was Headteacher at Norton College and other roles include Assistant Director for Education and Skills in both York and North Yorkshire as well as being the Director of Education with CfBT in Lincolnshire.
Jill has been a governor at Norton College since 2017 and a Trustee for Evolution since 2018. Since 2017, she has been a Trustee at South Bank Multi-Academy Trust in York, also undertaking the role of Vice Chair. She has chaired the Education committee at both Trusts.
Jill is delighted to be the Chair of Trustees for Evolution. She is passionate about Norton College, education and ensuring that all children and young people can have the best possible outcomes and life chances.
Karron Young
Karron has worked in education for over 30 years in a variety of roles and positions in the East Riding of Yorkshire and now at City of York Council. She is currently Head of the Virtual School and Inclusion Services Team.
Karron has been a governor at Norton College since 2012 and a Trustee of Evolution since 2016. She became Chair of the Local Governing Board in 2016 and has recently stepped down from this role to become Chair of the Education and Standards Committee.
Karron has three children who have all been educated at Norton College and is dedicated to her role as a Trustee to both support and challenge school leaders to continue improving the educational outcomes and life chances for all pupils.
Luke Ives
Information to follow
Craig Shepherdson
Craig was a student at Norton in the 1980's, before moving to Malton School sixth form, and then studying Electrical and Electronic Engineering at Leicester Polytechnic. After this he joined the Royal Air Force and enjoyed a full military career working initially in Communications and Electronics, and later in IT, Data Communications and Command & Control Systems.
After leaving the RAF, Craig and his family moved back to Norton, and he worked for IBM based in Farnborough. Since 2015 he has worked for an Austrian communications company, providing direct support to the MOD.
He became a governor at Norton in 2017 and has been Vice Chair of Governors/Trustees since 2018. Craig is parent appointed and provides a strong link between the College and the local community.
Samuel Moore
Sam has been a governor/trustee since 2020. He is a Chartered Tax Advisor who spent much of his career working in financial consultancy with PwC, including over 10 years spent in Sydney, Australia.
He currently runs his own consultancy based near Malton. Sam grew up near York and on his return to Yorkshire from Australia was very keen to get involved with education-based volunteer work.
Although he has no previous links to Norton College, he is delighted to be part of the governance team.
Helen Robinson
Kelly Dunn
Kelly is the founder and managing director of KD Recruitment who are an award-winning recruitment agency in Scarborough. Working in partnership with organisations across North and East Yorkshire, she supports those looking for careers within a number of industries and is passionate about changing people's lives.
Kelly attended Norton College herself and has been a trustee since December 2020. She loves to inspire the next generation and is also on the board of directors for NYBEP, supporting students with their careers after education. She is experienced in delivering workshops, careers fairs, speaking at events and encouraging children with mock interviews.
Caitlin Mackinder
Caitlin currently works for the Education Endowment Foundation leading their evidence mobilisation and school partnership work in Yorkshire and the North East of England.
Prior to this she worked for the Institute for Effective Education at the University of York, and for Teach for America where she led a large teacher preparation programme. Caitlin has also been a teacher and Head of English and has taught in schools with high proportions of disadvantaged pupils both in the UK and the USA. She is particularly interested in how schools use evidence to support disadvantaged and SEND students to overcome barriers to learning, and she is delighted to have the opportunity to champion these, and other groups, on the board of trustees for Norton College.
Catriona Stewart
Catriona's background in education consists of a MA in English from Oxford University, and 15 years teaching in a variety of posts in state and private schools, (which included working as a coach for Oxbridge entrance exams).
Since then, for twenty-five years, she has run her own international design business, selling her own products. Fifteen years ago she moved to Norton, and since then she has worked as a professional artist, with a studio one mile away from the town.
She is honoured to have been a governor at Norton College since 2016 and is committed to her role as a Trustee. She brings an outside view, and voice, to Norton College, and is dedicated to the school and the children who go there.
Trust Board (Members)
Trust Board Agendas and Minutes
Trust Board Documentation


Before the immigration crisis is over in Texas, hundreds of thousands of illegal aliens, many of them minors, will have entered the U.S. illegally from Mexico this year. A large portion of the illegal aliens are from El Salvador, Guatemala and Honduras. The Obama Administration has blamed the overwhelming influx on gang violence and poverty. Hogwash. The living conditions in the source countries have been stable for years. There is certainly violence and abject poverty in Central America, but there has been no change in living conditions that would prompt an unprecedented exodus of children.
Obama's steadfast refusal to enforce our immigration laws is directly responsible for the current "humanitarian crisis." Word has spread throughout this hemisphere that America has laid a welcome mat along our southern border. When illegal alien minors are taken into custody as they enter the U.S., they are not sent back to their home country. Obama's DHS simply flies or buses them to whatever city they were headed for. Once at their final destination, Obama's DHS turns them over to a relative or a friend who may or may not be an illegal alien. The illegal alien minor, as well as the person or family taking responsibility for the minor's care, stand virtually no chance of being removed from this country as required by law.
The wave of illegal alien children and families was a predictable result of Obama's lawless policy. Anyone with a rudimentary understanding of the dynamics of illegal immigration could have predicted this mess. In fact, many of us did. Hundreds of thousands of Central American aliens are now in the pipeline and we should expect their arrival over the next year.
Obama could shut the flow down by simply enforcing laws that are already on the books. As the situation stands now, illegal aliens seek out a Border Patrol agent when they cross the border. They understand that being taken into custody now means amnesty, not deportation. Instead of reinstating historic policies that prevented this sort of "humanitarian crisis," Obama has asked Congress for two billion dollars to pay for his folly. Nearly 40 percent of an agent's time is now spent processing children who will undoubtedly remain in the United States for the rest of their lives. The two billion dollars Obama is seeking will only pay for a fraction of the eventual cost to taxpayers.
During my career, I witnessed my share of boneheaded immigration policies that did more harm than good to our country. Obama has taken bad governance to a level not seen in decades.


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CDC - Cancer Cluster archived website note
This document is provided by the National Center for Environmental Health ONLY as an historical reference for the public health community. It is no longer being maintained and the data it contains may no longer be current and/or accurate.
For current information about cancer clusters, please visit
http://www.cdc.gov/nceh/clusters/
CDC's Role in Investigating Cancer Clusters
From 1961 to 1982, CDC investigated 108 reported cancer clusters in 29 states and 5 foreign countries (Caldwell GG. Twenty-two years of cancer cluster investigations at the Centers for Disease Control. Am J Epidemiol 1990; 132[1]:S43-S47). The studies were begun in hopes of identifying a viral cause of cancer clusters. During these investigations, however, no clear cause was determined for any of the reported clusters.
A separate CDC investigation of a non-work-related cluster of Kaposi's sarcoma in the early 1980s did lead to the discovery of a cause. It was found that the vast majority of Kaposi's sarcoma cases developed in association with human immunodeficiency virus (HIV) infection and acquired immunodeficiency syndrome (AIDS).
On limited occasions, as illustrated by the following examples, CDC works with specific state health departments to address public health concerns about potential cancer clusters:
Fallon, Nevada Exposure Assessment
Detailed description of the CDC investigation into a significant increase in number of children diagnosed with acute lymphocytic leukemia and acute mylocytic leukemia in Churchill County, Nevada in February 2001 ...more
Sierra Vista, Arizona Childhood Leukemia Investigation
Description and results of the CDC investigation into an elevated number of cases of childhood leukemia (acute lymphocytic leukemia [ALL] and acute myelocytic leukemia [AML]) in Sierra Vista, Arizona. ...more
Toms River, New Jersey Childhood Cancer Incidence Update (January 2003) [external link] [PDF, 103 Kb]
Updated report on the investigation conducted by the New Jersey Department of Health, in cooperation with the CDC, into a higher than expected number of cases of childhood cancer in Dover County, New Jersey.
Childhood Cancer Incidence Summary (September 1997) [external link] [PDF, 154 Kb]
Original detailed description of the investigation conducted by the New Jersey Department of Health, in cooperation with the CDC, into a higher than expected number of cases of childhood cancer in Dover County, New Jersey.
Woburn, Massachusetts Childhood Leukemia Study [external link] [PDF, 104 Kb]
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The next day (Monday) on my way to work, I stopped at Home Depot to look at pond liners. They didn't have any, but they did have some really nice half wine barrels made from oak. All the staves were nice and tight, so I bought one and put it in the back of my truck. When I got home, I didn't say anything to my wife about it, knowing that the next 2 days she'd be working. When I got home Tuesday afternoon, I put it in my back yard and filled it with water. There was major leaking between the staves, but I didn't care because I was planning on either sealing it with epoxy or getting a liner. After letting the water run into it for 3 hours, it would only fill to about 4 inches below the rim. My wife would be home soon, so I watered the garden and went inside.
The next morning, enough water had leaked out that the barrel was only about half full. OK, that's what I expected. I didn't try to fill it because I didn't want to wake up my wife and have her come outside to see what I was doing. When I came home for lunch (after she'd left for work), I filled the barrel again. When I got home from work that evening, some water had drained out so I filled it again and watched. This time I could get it to within 3 inches of the rim. I watered the garden and went inside to get the kitchen cleaned before my wife got home from work.
When I checked our pond Friday morning, the water was within 6 inches of the rim. I filled it and it went a little higher than the first time but I was in a hurry so din't have time to measure. I also thought I saw some movement under the water but I didn't have time to check it out. When I got home Friday evening I filled it again, and it went all the way to an inch from the rim. After it settled down, I decided to look for movement under the water and sure enough it was there – a tiny bird with wide wings flying around under water – kind of reminded me of a killdeer. Of course what it really is, is a backswimmer. The thing I don't understand is how did he get into my pond? He didn't come with the barrel (it was dry), and I doubt he came through the hose – no way he would have fit through the sprayer I use. So where did he come from and what is he eating?
After a week, my little pet is still in my pond. He's not very active during the day but in the evening he likes to swim around. Lots of fun to watch and I'm grateful to have him. I just wish I could figure out how he got here.
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So How did the night come about?
The night came about very naturally, I was already involved at Corsica with its 'genre smashing' night, Trouble Vision.
The club was crying out for a more focussed night for people just that little bit more serious about music, the team at Corsica are very supportive and it is the perfect club for me, size, sound, people. There are hardly any other places id rather put parties on in London to be honest!
You've been busy from the very first parties and there is always such a great vibe at CS, why do you think this is?
I think luck had a lot to do with it, timing was pretty essential a lot more people getting in to house and techno.
I made a few perhaps lucky bookings, getting artists like Bicep, Tornado Wallace and Mano le tough over for some of their first London gigs who have of course since blown up.
Our line ups have always been particularly big for the size of the club too, and I try and make the combinations of artists as interesting and as unique as possible.
Who does the programming and is it difficult to book the artists that you think combine well individually to play on the same night?
I program it all myself really, and yes, it is becoming increasingly difficult to book artists in general in London as a lot of people want similar artists.
To get a line up that goes great together booked, it really is a combination of planning ahead, good taste and a whole lot of luck! Of course the reputation at Corsica means booking artists is a lot easier I think as they generally want to come and play here.
There's been so many killer kine ups but which has been your favourite night so far?
Hmmm, I think maybe the one in February 2012 with Prosumer and San Soda in room 1 and Axel Boman and Midland in room 2, the rooms were balanced so well and both rooms offered something just a little bit different.
The Beats in space room in Feb 2013 with Todd Terje, Tim Sweeney and Casino Times was also a bit of a dream.
I know there are some festival gigs in the pipeline can you tell us a bit more about these?
Sure! We are doing a stage at Garden festival with myself, Ben Sun, Casino Times and Bicep which is going to be killer.
We are also running a boat party with Midland, Casino and myself at Dimensions festival and are presenting a stage at Gottwood festival in Wales with Wolf Music.
You recently mastered your first two EP's, which artists feature, when are these on release and how many copies are being pressed?
So the 2 Eps are from artist who have never released music (under these names) before, so im really happy about that.
The first (tief001) is by 'Sisterhood' which i am really amazed at how good it is. The two originals are remixed by Juju & Jordash and Arttu which raise the profile of the release in general and compliment the originals perfectly.
The second is by a guy named Esa who i met in Cape Town and Ooft! who produce together as 'Prophets of the south'.
Esa produces as himself and is also involved with Auntie Flo and Ooft! of course is a well established producer in his own right.
The pairing is great and the 3 tracks they gave me are outstanding, really nice african sounding and almost ravey house.
I am pressing 300 of each to begin with, but im sure the demand will be there so will probably up to 500, we arent bothering with all that 'limited numbered' stuff.
You've already booked so many outstanding artists is there anyone left on your hit list who would be a dream to book?
I have actually already covered a lot of dream bookings to be honest!
Pepe Bradock being one of them (who will be playing at our birthday in September).
Id love to book Thomas Bangalter of Daft Punk, but realise that this is pretty much impossible..
Ah, Dj Harvey, yeah he is up there. Again is a really long shot but would be killer!
Most club nights have a shelf life, would you ever consider killing off Tief at the height of its popularity to maintain the memories if how special it was?
Wow thats a depressing thought! Its not something i would do any time soon at all, as romantic that idea is. There are plenty of new amazing artists and sounds emerging that i am becoming really interested in so im going to have to say no!
I am REALLY glad you said that!


Yesterday was the second installment of Jonah at North Ridge. We discussed chapter 2 where Jonah turns and faces the task God is trusting him with; undoubtedly nervous, probably skeptical, but decidedly faithful.
Before reading further, take a look at last week's representation of Jonah 1, 'The Word of the Lord Came to Jonah'. In last week's image, the page was mostly black. A small circle of light was growing in the bottom right corner, symbolizing the overwhelming wave of fear that came over Jonah once he realized the task before him. This week takes that image a step further in the story. The white circle is still present in the bottom right, still being restrained by the overwhelming darkness of the background. A face is protruding from the dark background, symbolizing Jonah turning to face his fear and tackle his mission head on. We're required to do the same thing in our lives. We first turn from our sinful, selfish and materialistic selves to embrace God. Then we have multiple occasions when God will give us a specific task or bring us to a crossroad where we must choose again, our way or God's way. As Jonah, among many others, found out, God's way is always best.
Look back at last week's drawing and then this one again. I proposed the emotion I conveyed last week was like the small white circle being told to make the page white again. The outer perimeter is encroaching on the darkness, reducing the space it holds on the page. I used shapes to mimic that of ocean waves and seaweed that swept over Jonah and wrapped around his legs during his time in the ocean.
As this series progresses, I do not believe we will be seeing a separate drawing for each chapter, I am expecting this will become a single image that morphs from fear, to repentance, to victory, then on to resentment and back again. Expect something brighter for next week, Jonah 3!


text:
Expecting mother Rachel Dolezal to name her son after Harlem Renaissance poet Langston Hughes
By Tobias Salinger
Rachel Dolezal previously said during an interview on the "Today" show that she identifies as "transracial." She revealed she plans to name her son after the poet Langston Hughes in an interview with Vice on Monday. (HANDOUT/Reuters/TODAY/Anthony Quintano)
Outed white woman Rachel Dolezal said she plans to name her son after African-American icon Langston Hughes, according to a report.
Dolezal, 38, revealed in August that she had gotten pregnant amid the outcry over her claims that she is black. She resigned from her position as the president of the Spokane, Wash., chapter of the NAACP after a Spokane TV station exposed her as the daughter of a white couple in June.
Dolezal, who now works as a hairdresser, declined to discuss the expected baby's father but explained her choice of names in an interview with Vice's female-focused news site Broadlyon Monday.
BEST REACTIONS TO KIMYE'S NEW BABY'S NAME, SAINT
"I'm naming him Langston because of Langston Hughes' 'Mother to Son' poem," Dolezal said. "Life hasn't been easy for me at all, but I keep going. I'm still climbing, so don't you sit down and stop. You keep going, and I want that to be a lesson for all my sons."
Rachel Dolezal poses in a March 2015 file photo. A Spokane, Wash., TV station outed her as the daughter of white parents in June. (Colin Mulvany/AP)
Dolezal keeps Hughes' "Collected Poems" on a chest in her living room, according to Vice. The Harlem Renaissance legend and outspoken civil rights activist's poem "Mother to Son" is one of Hughes' best known works. The mother's trials serve as a symbol of African-Americans' struggle for equality in the poem.
"Well, son, I'll tell you: life for me ain't been no crystal stair," the poem reads. "Don't you set down on the steps 'cause you finds it's kinder hard. Don't you fall now -- for I'se still goin', honey, I'se still climbin', and life for me ain't been no crystal stair."
A February 1959 portrait shows the poet, playwright and civil rights activist Langston Hughes eight years before his death. Outed white woman Rachel Dolezal said she is naming her expected son "Langston" after one of Hughes' best-known poems. (Underwood Archives/Getty Images)
The flak over Dolezal's race has prompted references to Hughes' works before. North Carolina State University historian Blair L.M. Kelley mentioned the short story "Who's Passing for Who?" while slamming Dolezal in an op-ed for The Washington Post in June. The Harlem-based story depicts a man and woman who fool both white and black people about their races until no one can say for sure whether the couple is white or black.
Kelley brought up the story to refute those who compared Dolezal to light-skinned African-Americans said to have "passed" as white during the Jim Crow era and the fight for black civil rights, the professor wrote.
This deleted Instagram post from August shows Dolezal's announcement of her pregnancy. (Instagram)
"There may have been a role for Dolezal to play as ally in that movement if she had pursued it — but not as a white woman masquerading as a black woman," Kelley wrote. "Absent the painful choice between second-class citizenship and a lie — and absent the threat of repercussions beyond embarrassment — her elaborate performance needs another name."
Dolezal defined herself as "transracial" in a silence-breaking interview on NBC's "Today" show. She has one biological son who is a teenager and a 21-year-old brother she adopted.
Dolezal told Vice a third of her friends no longer talk to her and choked up while recalling her birth in a teepee in Montana and her strict religious upbringing.
"I've struggled with depression to the point where I wonder if it's even in my kids' best interest for me to stay around," Dolezal said. "I feel like I'm a liability to my own children."
Follow on Twitter @tobysalkc
tsalinger@nydailynews.com


text:
KBS Representatives Give Statement Regarding Broadcast Of NCT Jaehyun's K-Drama, "Dear M" In 2022
Here's the entire timeline of the drama's multiple postponements.
Previously, it was reported that NCT's Jaehyun would be making his first acting debut with K-Drama, Dear M. Many fans had been ardently looking forward to the drama, given that it would be his first challenge at acting and at a romantic role.
Originally filmed at teased in late 2020, the drama dropped it's first teaser in January 2021. It was slated for an air in February 2021, coinciding with the Valentine's day season. However, it was pushed back due to co-star Park Hye Su's bullying controversy.
Dear M tried to secure a later airing slot in August 2021, but with the continuing investigations for Park Hye Su's scandal, they had to shelve plans once again. Fans got excited when Dear M suddenly appeared on OTT platform, iQiyi's homepage. However, no news of the show actually airing was given.
Finally, as Park Hye Su's name was cleared by Dispatch, it seems that the drama might see the light again. However, the police investigations are still ongoing. But in 2022 January, KBS representatives gave a statement that disappointed fans once more.
Firstly, it's going to be hard to air it in the first half of 2022. The results of the police investigation needs to be out for us to discuss the broadcast.
— KBS Representative
As such, it is likely that the show will be postponed once more, even over a year since filming. Will it ever be aired?
Dear M is a romance drama about college students. Jaehyun plays Cha Minho, a handsome and studious student who is best friends with Ma Joo A, played by Park Hye Su. One day, the campus is turned upside down with an anonymous confession on a social media page.
Source: Star MT
Here Are 6 Reasons Why NCT's Jungwoo Is A True Ace
Super Junior's Eunhyuk Had The Funniest First Reaction After Being Introduced To The SM Culture Universe
NCT's Mark Lee Is Getting His Long-Awaited Solo Debut — And NCTzens Are Definitely Not Okay
NCT DREAM's Jeno And Haechan Were Caught Off Guard By A Clothing Mishap
See more NCT


Down the Mine is the twenty-fifth episode of the first season. One day while Thomas is waiting at the junction, Gordon, who is in disgrace after recently causing trouble by falling into a ditch, pulls in with some trucks.
I read somewhere recently that super elevation – the banking of track on the curves – was difficult to achieve on a model railway and not really worth attempting.
Really lovely work Dave. The soldering is really fine and it does look like a nice kit to build. Look forwards to the next thrilling instalment!
Here we see the model Gas Works from the "business end". The details of which bit does what are contained in the key below the plan.
Michael got in touch. Just look at his railroad. Superb. Best of all are the construction photos at the end – you can see how it did it from start to finish. Just superb isn't it?
Bognor Regis Model Railway Club's and Members' Layouts. Club Layouts. Here you can view pictures and details of our club layouts. If you would like one of these to attend your exhibition, please contact our exhibition manager via the Contact Us page.


Sidney Volunteer Firefighters enter a house in the 2100 block of 22nd Ave. after responding to a report of smoke coming out of the back of the house. Firefighters extinguished the fire after about 20 minutes.
Sidney's Volunteer Fire Department responded to a house fire late Friday morning in the 2100 block of 22nd Ave. in Sidney.
Black smoke poured from the back of the home when responders arrived to the scene. At one point, fire spewed out of a window near the back of the home during the fire department's attack on the fire. It took firefighters about 20 minutes to bring the blaze under control as neighbors watched.
The home was unoccupied at the time of the fire, and no one was injured. Kristopher Gadway, head of the household of five, was at work near Kimball, while other family members were in Sterling.
"I just thank the Lord no one was here or hurt," Gadway said as firemen mopped up the scene.
Sidney Fire Chief LaVerne Bown said the state fire marshal did not make a determination on the cause of the blaze, which would be ruled accidental. The house was deemed a total loss.
Sidney firefighters responded with three vehicles and 16 firefighters. Also responding were the Sidney Police Department and Regional West Ambulance.


Test weight and grain quality have been a popular topic of discussion on social media lately. At Hoegemeyer, we take great discussions like this to heart and want to provide information on test weight in corn to use in your next discussion on this topic.
Grain yield and kernel test weight are typical parts of the cafe discussion around hybrid selection in the winter or corn harvest in the fall. Test weight also factors into the price at the elevator when discounts are applied for delivered grain that weighs less than 56 lb/bu. While the terms bushels and test weight are bantered around, it might be worthwhile to dig into these terms and compare their importance at the elevator compared to the feed mill (or total mixed ration mixer).
Test weight helps account for the varying densities of grain caused by weather and/or production practices. There are many causes of reduced kernel fill including hybrid genetics, low temperatures, reduced solar radiation (cloudy conditions), premature death due to leaf disease, ear rots, drought and freeze damage to late-developing fields.
When kernels are prevented from filling completely and/or maturing and drying naturally in the field, the starch molecules inside the grain are prevented from the natural process of shedding absorbed water molecules that allows the grain to shrink to a normal size. Artificial drying with heat removes this excess water but the starch molecules do not shrink, and grain size doesn't change appreciably, so test weight remains relatively low.
In some situations, mature kernels significantly hydrated by rain, dew or fog can initiate early germination in the field during which starch is utilized for energy leaving small voids inside the grain. Although the grain may again dry in the field, the seed size does not change, and the small voids inside the seed can result in a decreased test weight.
Test weight for a given hybrid can vary from field to field and from year to year and does not correspond very well to the yield potential of the environment. Data from Purdue University shows the lack of correlation between grain yield and test weight in nitrogen rate trials over multiple site-years in Indiana.
At the elevator, 1 bu. of corn is defined as 56 lb. independent of test weight. The weight of the corn for which a seller is paid is, therefore, not affected by test weight - 56 lb. is 56 lb. However, the price the seller receives for a bushel can be affected by test weight. Transporting and storing lower bulk-density grain is more expensive (on a weight basis), so buyers discount grain on a sliding scale if test weight is below minimum standards.
Higher weight per unit volume of grain has obvious advantages in harvest, handling and storage: More bushels will fit in the combine grain tank, on a truck or in a grain bin.
Kernel moisture and test weight are inversely related with higher moisture kernels displaying lower test weight. As corn dries in the field or in a dryer, test weight naturally increases if kernel integrity remains intact. Test weight increases because dried kernels shrink and are slicker so tend to pack more tightly in a quart cup. Kernels are also denser than water, so bulk density goes up as water exits the kernel.
While it is true that test weight will increase due to natural or artificial drying, the increased test weight means the corn will occupy less total volume because of increasing the corn bulk density and could potentially reduce test weight dockage at the elevator. It does not mean there will be any more bushels (pounds) to sell because 1 bu. is 56 lb. regardless of test weight.
When the growing season produces low-test corn, it is important to feed corn based on weight and nutrient content and not volume such as bucket loads.
Research on feeding light-test weight corn to dairy cattle is lacking. However, there is a considerable body of research from beef cattle showing that moderately low-test corn (46-48 lb/bu.) results in similar daily gain and feed efficiency to normal corn. This supports an early study from Minnesota showing very little difference in the gross energy of corn ranging from 35 lb to 56 lb/bu.
Due to a lowered starch dilution effect, light-test corn will be higher in crude protein and fiber, so analyzing nutrient content and using modern summative net energy equations may be the best way to determine feeding rates.
Research supports the sage advice of selling high-test weight corn and feeding low-test corn. While test weight, a measure of grain bulk density, has important implications for pricing at the elevator and transportation/storage costs, it has less significance for feeding the grain to ruminants.
The primary watch-outs for feeding light-test corn include a tendency for faster ruminal starch digestion rates, monitoring for mycotoxin levels, the need to feed by weight not volume and close attention to processing consistency.


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Before releasing a new version with the MergeAction bug fixed I'd like to write a test case for this. And before writing that test case I'd like to convert the current testing framework (which uses fablib) to a normal one (callable using setup.py test ).
For Lino, this conversion is now done.
New module lino.projects.cosi.tests.cosi_tests does already the preliminary stuff.
The new project atelier contains those parts of djangosite which are useful also without Django. But atelier.test.SubProcessTestCase is new.


Helping people deal with debt is just one of the services offered by Grampian Housing Association's (GHA) SMART Financial Inclusion and Early Intervention Project. With the ability to offer advice and support on debt management to over 8,000 people across Aberdeen, Aberdeenshire and Moray the service is helping 100s of people across the region.
David Wiseman, 62, from Aberdeen was in dire straights when he was referred to the GHA's project by his housing officer. His financial worries had begun a decade earlier and despite current support he is still working out his money worries.
David says, "It all kicked off about ten years ago. I had gone through a divorce and I was drinking quite a bit at the time. But the one thing that really led me spiral into debt were my council tax repayments. The council tax office had me under a different first name and I told them wasn't paying up until it was in the correct name. But the thing was I didn't put the money aside I just drank it.


text:
package alma.acs.monitoring;

import java.lang.management.ManagementFactory;
import java.net.InetAddress;
import java.net.UnknownHostException;
import java.util.List;

import junit.framework.TestCase;

/**
 * @author rtobar
 * created Feb 18, 2008 2:06:46 PM
 */
public class RemoteThreadsClientTest extends TestCase {

	private RemoteThreadsClient rtc = null;

	protected void setUp() throws Exception {
		super.setUp();

	}

	protected void tearDown() throws Exception {
		super.tearDown();
	}

	/*
	 * Test method for 'alma.acs.monitoring.RemoteThreadsClient.RemoteThreadsClient(String)'
	 */
	public void testRemoteThreadsClientString() {
		boolean throwedException = false;

		// We know that java.lang.String has now main method!
		try {
			rtc = new RemoteThreadsClient("java.lang.String");
		} catch(RemoteThreadsException e) {
			throwedException = true;
		}

		assertTrue("Can't connect to java.lang.String",throwedException);

		// Let's try connecting to the class ""
		throwedException = false;
		try {
			rtc = new RemoteThreadsClient("");
		} catch(RemoteThreadsException e) {
			throwedException = true;
		}

		assertTrue("Can't connect to the \"\" class",throwedException);

		List<String> lala = ManagementFactory.getRuntimeMXBean().getInputArguments();
		for(int i =0; i!= lala.size(); i++)
			System.out.println(lala.get(i));
	}

	/*
	 * Test method for 'alma.acs.monitoring.RemoteThreadsClient.RemoteThreadsClient(int)'
	 */
	public void testRemoteThreadsClientInt() {

		boolean throwedException = false;
		try {
			rtc = new RemoteThreadsClient(0);
		} catch (RemoteThreadsException e) {
			throwedException = true;
		}
		assertTrue("PID can't be 0", throwedException);

		throwedException = false;
		try {
			rtc = new RemoteThreadsClient(
					Integer.valueOf(
							ManagementFactory.getRuntimeMXBean().getName().split("@")[0]
					));
		} catch (RemoteThreadsException e) {
			throwedException = true;
		}

		assertTrue("PID of the current application", !throwedException );
		assertNotNull(rtc);
	}

	/*
	 * Test method for 'alma.acs.monitoring.RemoteThreadsClient.RemoteThreadsClient(InetAddress)'
	 */
	public void testRemoteThreadsClientInetAddress() {

		rtc = null;
		boolean remoteThreadsException = false;
		boolean unknownHostException   = false;

		try {
			InetAddress localhost = InetAddress.getByName("localhost");
			rtc = new RemoteThreadsClient(localhost);
		} catch (RemoteThreadsException e) {
			remoteThreadsException = true;
		} catch (UnknownHostException e) {
			unknownHostException = true;
		}

		assertTrue(!remoteThreadsException);
		assertTrue(!unknownHostException);
		assertNotNull(rtc);
	}

	/*
	 * Test method for 'alma.acs.monitoring.RemoteThreadsClient.RemoteThreadsClient(InetAddress, int)'
	 */
	public void testRemoteThreadsClientInetAddressInt() {

		rtc = null;
		boolean remoteThreadsException = false;

		// Let's try with a negative port...
		try {
			InetAddress localhost = InetAddress.getByName("localhost");
			rtc = new RemoteThreadsClient(localhost,-12);
		} catch (RemoteThreadsException e) {
			remoteThreadsException = true;
		} catch (UnknownHostException e) {
		}

		assertTrue(remoteThreadsException);
		assertNull(rtc);

		// Now with a good port...
		remoteThreadsException = false;
		try {
			InetAddress localhost = InetAddress.getByName("localhost");
			rtc = new RemoteThreadsClient(localhost,1234);
		} catch (RemoteThreadsException e) {
			remoteThreadsException = true;
		} catch (UnknownHostException e) {
		}

		assertTrue(!remoteThreadsException);
		assertNotNull(rtc);
	}

	/*
	 * Test method for 'alma.acs.monitoring.RemoteThreadsClient.connect()'
	 */
	public void testConnect() {

		boolean throwedException = false;

		rtc = null;

		// Connect to this process and get the mbean
		try {
			rtc = new RemoteThreadsClient(
					Integer.valueOf(
							ManagementFactory.getRuntimeMXBean().getName().split("@")[0]
					));
		} catch (RemoteThreadsException e) {
			throwedException = true;
		}

		assertTrue("Connection to the MBean unsucessful",rtc.connect());
		assertTrue(!throwedException);
	}

	/*
	 * Test method for 'alma.acs.monitoring.RemoteThreadsClient.getMBean()'
	 */
	public void testGetMBean() {

		RemoteThreadsMBean mbean = null;
		boolean throwedException = false;

		rtc = null;

		// Connect to this process and get the mbean
		try {
			rtc = new RemoteThreadsClient(
					Integer.valueOf(
							ManagementFactory.getRuntimeMXBean().getName().split("@")[0]
					));
			rtc.connect();
			mbean = rtc.getMBean();
			rtc.close();
		} catch (RemoteThreadsException e) {
			throwedException = true;
		}

		assertNotNull(rtc);
		assertNotNull(mbean);
		assertTrue(!throwedException);
	}

	/*
	 * Test method for 'alma.acs.monitoring.RemoteThreadsClient.close()'
	 */
	public void testClose() {

		boolean throwedException = false;
		rtc = null;

		// Create the client and close it (should throw exception)
		try {
			rtc = new RemoteThreadsClient(
					Integer.valueOf(
							ManagementFactory.getRuntimeMXBean().getName().split("@")[0]
					));
			rtc.close();
		} catch (RemoteThreadsException e) {
			throwedException = true;
		}

		assertTrue(throwedException);

		// Let's do it as it should have been done
		throwedException = false;
		try {
			rtc = new RemoteThreadsClient(
					Integer.valueOf(
							ManagementFactory.getRuntimeMXBean().getName().split("@")[0]
					));
			rtc.connect();
			rtc.close();
		} catch (RemoteThreadsException e) {
			throwedException = true;
		}

		assertTrue(!throwedException);
	}
}

redpajama_set_name: RedPajamaGithub


Miss Martin, 30, of Gondree, Carlton Colville, Lowestoft, has worked at Decoy Farm since it opened as a specialist nurse-led service for young adults with learning disabilities.
It was a 100th birthday wish that Joyce Fitzsimons could not have dreamed of coming true.
For a special guest at her party in Timperley Nursing Home today <February 15> was her best friend of 56 years, Diane Handiside, 77, who flew 12,000 miles from Christchurch, New Zealand, especially for the occasion.


text:
NetManage Picks Up Wall Data
Is there still a market for terminal-based PC-to-host connectivity solutions in the age of the Internet? NetManage Inc. (www.netmanage.com) certainly hopes that there is, especially after the PC connectivity specialist moved in late October to acquire PC terminal emulation software specialist Wall Data Inc. (www.walldata.com).
The companies signed a definitive agreement for NetManage to buy Wall Data for about $94 million. NetManage is offering to acquire all of Wall Data's outstanding shares at $9 per share.
Wall Data is best known for its Rumba brand of PC-to-host connectivity solutions, which are deployed primarily as terminal front ends for Unix and IBM-based AS/400 and System 390 mainframe computers. NetManage provides connectivity solutions for other hosts and platforms, as well.
The client-to-host connectivity space has suffered of late as more organizations opt to put Web browser front ends on host-based applications. These applications were traditionally accessed through a classic dumb terminal or through a PC equipped with terminal emulation software -- such as that produced by Wall Data.
"[In past surveys] IT decision-makers indicated that they planned to put Web-based front ends on these important applications rather than continuing to use terminal emulation software," says Dan Kusnetzky, director of worldwide operating environments at International Data Corp. (IDC, www.idc.com). "This means that these end-user organizations are planning to update their middleware -- transaction processing monitors, etc. -- to use the versions which support browser-based clients."
During much of 1999 Wall Data was besieged by a series of financial setbacks stemming from the shrinking demand for solutions in the client-to-host connectivity market segment. In August, for example, the company divested itself of a 10 percent holding in XML developer DataChannel (www.datachannel.com) to raise money.
Four days before the proposed acquisition by NetManage, Wall Data founder and CEO John Wall resigned. On the same day, the company announced an anticipated operating loss of between $11.5 million and $12.5 million for the first quarter of 2000.
NetManage, for its part, is recasting itself as a purveyor of e-business solutions. In March 1999, for example, NetManage unveiled its eN2000 strategy, an initiative to offer solutions for server-based connectivity, Web-to-host, and application integration server implementations. Consequently, company sources expect the combination of NetManage and Wall Data to heavily leverage technologies from both parts to assist enterprises that want to transition to e-business solutions.
NetManage also indicated that the combined entity will continue to provide client-to-host connectivity solutions such as those traditionally supplied by its constituent parts. In addition, sources say, NetManage hopes to increase its presence in the burgeoning Web-to-host connectivity market.
Zvi Alon, NetManage's president and CEO, says his company's acquisition of Wall Data was prompted by a desire to increase its presence in the explosive market for e-business solutions. "With the acquisition of Wall Data, we are continuing to further our strategy of market leadership while increasing our presence in the new and growing e-commerce and e-business markets," Alon comments. "This move will strengthen the position that we established as the first single source supplier of advanced PC and Web-based connectivity solutions."