text:
\section{Introduction }

As a fundamental model in the theory of integrable systems, the Kadomtsev--Petviashvili (KP) hierarchy can be defined as follows. Let
\begin{equation} \label{Lckp}
L_{KP}=\p+v_1\p^{-1}+v_2\p^{-2}+\dots, \quad \p=\frac{\od}{\od x},
\end{equation}
be a pseudo-differential operator whose coefficients $v_i$ are scalar unknown functions of the spacial coordinate $x$, then the KP hierarchy is composed by the following evolutionary equations of $v_i$ as
\begin{equation}\label{kp}
 \frac{\p {L_{KP}}}{\p t_k}=\left[({L_{KP}}^k)_+, {L_{KP}}\right],\quad
 k=1,2,3,\dots.
\end{equation}
Here and below the subscript ``$+$'' of a pseudo-differential operator means to take its purely differential part, while the subscript ``$-$'' means to take its negative part. Suppose that the equations \eqref{kp} are imposed with the constraint $({L_{KP}}^n)_-=0$ with some integer $n\ge2$, then they form the Gelfand--Dickey (or the $(n-1)$-th Korteweg--de Vries) hierarchy.

It is known that the KP hierarchy \eqref{kp} possesses a series of
bi-Hamiltonian structures derived by the $R$-matrix formalism
\cite{STS}, and that it can be represented into the form of bilinear
equation of (adjoint) Baker--Akhiezer functions or of a tau function
\cite{DKJM-KPBKP}. Such bi-Hamiltonian structures and the bilinear
equation can be reduced to that of the Gelfand--Dickey hierarchies.
What is more, for the KP hierarchy there is a class of
non-isospectral symmetries named as the additional symmetries that
can be constructed via the so-called Orlov--Schulman operators
\cite{OS}. The flows of such additional symmetries commute with all the
time flows $\p/\p t_k$ but do not commute between themselves;
instead, they generate the $W_{1+\infty}$ algebra. As an
application of the additional symmetries for the KP
hierarchy, part of these symmetries can be reduced to the Virasoro symmetries
for its subhierarchies such like the Gelfand--Dickey hierarchies. Note that Virasoro symmetries reveal crucial
properties of a large amount of integrable hierarchies including the
Gelfand--Dickey hierarchies and the Drinfeld--Sokolov hierarchies,
see e.g. \cite{AvM, Di-add1, Di-add2, Wu-DS} and references therein.

In the definition of the KP hierarchy \eqref{kp}, it is used pseudo-differential operators with only finitely many positive powers in $\p$.
The notion of pseudo-differential operator was generalized in \cite{LWZ} to be over a certain graded differential algebra $\cA$ such that these operators may contain infinitely many positive powers in $\p$ (see Section~2 below for details). By using these operators, in \cite{WZ} Zhou and one of the authors of the present paper considered an integrable hierarchy, which can be viewed as a subhierarchy of the dispersionful analogue \cite{SB} of the universal Whitham hierarchy. More exactly, let
\begin{equation} \label{PhP}
P= \p+\sum_{i\ge1}u_i  (\p-\varphi)^{-i}, \quad \hat{P}=
(\p-\varphi)^{-1}\hat{u}_{-1}+\sum_{i\ge0}\hat{u}_i (\p-\varphi)^i
\end{equation}
with $u_i, \hat{u}_i$ and $\varphi$ being certain unknown functions belong to $\cA$, then the
following evolutionary equations are well defined:
\begin{align}\label{ekp1}
& \frac{\p }{\p t_k}(P,\hat{P})=\left([(P^k)_+, P],[(P^k)_+,
\hat{P}]\right), \quad \frac{\p}{\p \hat{t}_k}(P,\hat{P})=\left([-(\hat{P}^k)_-, P],
[-(\hat{P}^k)_-, \hat{P}]\right),
\end{align}
where $k=1, 2, 3, \dots$. These evolutionary equations compose an integrable hierarchy, which is named as the extended KP hierarchy. In fact, this hierarchy is reduced to the KP hierarchy \eqref{kp} whenever $\hat{P}=0$ (note that the operator $P$ gives an alternative expression of $L_{KP}$). On the other hand, if one lets $\varphi\to0$ and impose certain B-type symmetry conditions to the operators $P$ and $\hat{P}$, then the flows in \eqref{ekp1} with $k\in\Z_{>0}^{\mathrm{odd}}$ give the two-component BKP (2-BKP) hierarchy \cite{DJKM-KPtype, LWZ}. With the $R$-matrix method applied in the cases of the KP and the 2-BKP hierarchies \cite{STS, Wu-Rmat}, the extended KP hierarchy \eqref{ekp1} was shown to possess infinitely many bi-Hamiltonian structures \cite{WZ}.

In this paper we assume $\varphi=\p(f)$ in \eqref{PhP} with some homogeneous function $f\in\cA$ of degree $0$. We will show that the operators $P$ and $\hat P$ can be represented in a dressing form as
\begin{equation} \label{PPh0}
P=\Phi \p\Phi^{-1},\quad \hat{P}=\hat{\Phi}
\p^{-1}\hat{\Phi}^{-1},
\end{equation}
where $\Phi$ and $\hat\Phi$ are pseudo-differential operators of the form
\begin{align*}
&\Phi=1+\sum_{i\ge 1}a_i \p^{-i},\quad
\hat{\Phi}=e^f \left(1+\sum_{i\ge 1}b_i \p^{i}\right).
\end{align*}
With the help of these two dressing operators in the extended KP hierarchy, we will introduce two Baker--Akhiezer functions $\psi(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)$, $\hat\psi(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)$ and their adjoints $\psi^\dag(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)$, ${\hat\psi}^\dag(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)$ that depend on the time variables $\bm{t}=(t_1,t_2,t_3,\dots)$,
$\hat{\bm{t}}=(\hat{t}_1,\hat{t}_2,\hat{t}_3,\dots)$, and a nonzero parameter $z$. Our first main result is the following theorem (see Theorem~\ref{thm-bl} for a more precise version).
\begin{thm}\label{thm-bl0}
The extended KP hierarchy \eqref{ekp1} can be represented equivalently to a bilinear equation as
\begin{equation}\label{bl}
\mathrm{res}\,_{\!z}\left(\psi(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)\psi^\dag(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}', \hbt'; z)\right)=\mathrm{res}\,_{\!z}\left(\hat{\psi}(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)\hat{\psi}^\dag(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}', \hbt'; z)\right)
\end{equation}
with arbitrary time variables $(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt)$ and $(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}', \hbt')$.
\end{thm}

Our second main result is the construction of  additional symmetries
for the extended KP hierarchy, with the help of  certain
Orlov--Schulman operators given by the above dressing operators
$\Phi$ and $\hat\Phi$. Such additional symmetries will be
shown to generate a $W_{1+\infty}\times W_{1+\infty}$ algebra. These
results will be applied to study the $(n,1)$-constrained KP
hierarchy (see, e.g., \cite{ANP, BX, Cheng-ncKP,
Di, Kr}), which is denoted as $\hbox{cKP}_{n,1}$ and can be reduced from the extended KP hierarchy \eqref{ekp1}
under the constraint
\[
P^n=\hat{P}
\]
with a given integer $n\ge1$. In fact, the Virasoro symmetries for the  $\hbox{cKP}_{n,1}$ hierarchy (even for more general cases) were proposed by Aratyn, Nissimov and Pacheva \cite{ANP}, with the method of adding certain ``ghost'' symmetry flows such that some nonlocal actions on functions are involved. They also showed that, the  $\hbox{cKP}_{n,1}$ hierarchy subject to the subsidiary condition of invariance under the lowest Virasoro symmetry flow can be applied to compute explicit Wronskian solution for the two-matrix model partition function  \cite{ANP-DB}. As to be seen, the Virasoro symmetries appearing in \cite{ANP, ANP-DB} for the  $\hbox{cKP}_{n,1}$ hierarchy can be obtained in an alternative way, starting from the additional symmetries for the extended KP hierarchy (see Proposition~\ref{thm-ANP} below).

This paper is arranged as follows. In the next section, we will recall the pseudo-differential operators of the first and the second types over a graded differential algebra. In Section~3, we will recall the definition of the extended KP hierarchy, and represent it into the form of a bilinear equation. In Section~4, we will construct the additional symmetries for the extended KP hierarchy, and then study the Virasoro symmetries for the $\hbox{cKP}_{n,1}$ hierarchy. The final section is devoted to some remarks.

\section{Pseudo-differential operators}\label{sec-eKP}

Let $\cA$ be a commutative associative algebra, and $\p: \cA\to\cA$
be a derivation. We consider the linear space $\cD(\cA)=\set{\sum_{i\in\Z}f_i\p^i\mid f_i\in\cA}$ and its subsets. For instance, the set of pseudo-differential operators is
\[
\cA((\p^{-1}))=\set{\sum_{i\le k}f_i\p^i\mid f_i\in\cA, k\in\Z},
\]
and it becomes an associative algebra if a product is defined by
\begin{equation}\label{pro}
f \p^i\cdot g \p^j=\sum_{r\geq0}\binom{i}{r}f\, \p^r(g)\,
\p^{i+j-r}, \quad f,\,g\in\cA.
\end{equation}
For any two  pseudo-differential operators $A$ and $B$, their
commutator means $[A,B]=A B-B A$. Clearly, one has $[\p, f]=\p(f)$
for any $f\in\cA$.

In the present paper we assume the algebra $\cA$ to be a graded one.
Namely, $\mathcal{A}=\prod_{i\geq0}\cA_i$, such that
\[
\cA_i\cdot\cA_j\subset\cA_{i+j}, \quad \p(\cA_i)\subset\cA_{i+1}.
\]
Denote $\cD^-=\cA((\p^{-1}))$, which is called the algebra of
pseudo-differential operators of the first type over $\cA$. In
contrast, by the algebra of pseudo-differential operators of the
second type over $\cA$ it means \cite{LWZ}
\begin{equation}\label{Dpl}
\cD^+=\left\{ \sum_{i\in\Z}\sum_{j\ge \max\{0,k-i\}}a_{i,j} \p^i
\mid a_{i,j}\in\cA_j, k\in\Z \right\},
\end{equation}
whose product is also defined by \eqref{pro}. One observes that an
operator in $\cD^+$ may contain infinitely many positive powers in
$\p$.

Given an element $A=\sum_{i\in\Z} {f}_i \p^i\in\cD(\cA)$, its differential part, negative part and residue are defined
respectively as:
\begin{equation}\label{}
A_+=\sum_{i\ge0} f_i \p^i, \quad A_-=\sum_{i<0} f_i \p^i, \quad \mathrm{res}\,
A=f_{-1}.
\end{equation}
It is easy to see that
\[
\left(\cD^\mp\right)_\pm\subset \cD^-\cap\cD^+.
\]
What is more, on each $\cD^\mp$ there is an anti-automorphism
defined by
\[
\p^*=-\p, \quad f^*=f \hbox{ with } f\in\cA.
\]
Clearly, for any $A\in\cD^\mp$, one has
\[
\mathrm{res}\, A^*=-\mathrm{res}\, A.
\]

In what follows we will use the notation $\cA_{\ge r}=\prod_{i\ge
r}\cA_i$ with $r\in\Z_{\ge0}$. Given an element $\varphi\in\cA_{\ge1}$,
the following two maps are well defined with $\p$ replaced by
$\p-\varphi$, that is,
\begin{align}
\mathscr{S}_\varphi: \quad \cD^\mp&\to\cD^\mp, \nonumber\\
 \sum {f}_i \p^i &\mapsto \sum {f}_i (\p-\varphi)^i.
\label{Svp}
\end{align}
For instance, we have
\begin{align}\label{}
\mathscr{S}_\varphi(\p^{-1})=&(\p-\varphi)^{-1}=\p^{-1}(1-\varphi\p^{-1})^{-1} \nonumber\\
=&\p^{-1}+\p^{-1}\varphi\p^{-1} +\p^{-1}\varphi\p^{-1}\varphi\p^{-1}+\dots.
\end{align}
In \cite{WZ}, it was verified that the maps $\mathscr{S}_\varphi$ are
automorphisms on each $\cD^\mp$. Accordingly, the algebras $\cD^\mp$
can be represented as follows:
\begin{align}\label{Dvpm}
\cD^-=&\set{\sum_{i\le k}g_i(\p-\varphi)^i\mid g_i\in\cA, k\in\Z},\\
 \cD^+=&\left\{
\sum_{i\in\Z}\sum_{j\ge \max\{0,k-i\}}b_{i,j} (\p-\varphi)^i \mid
b_{i,j}\in\cA_j, k\in\Z \right\}, \label{Dvpp}
\end{align}
and their product can be defined equivalently by
\[
f (\p-\varphi)^i\cdot g (\p-\varphi)^j=\sum_{r\geq0}\binom{i}{r}f\,
\p^r(g)\, (\p-\varphi)^{i+j-r}, \quad f,\,g\in\cA.
\]
Moreover, it is easy to verify the following properties: for any
$A\in\cD^\mp$,
\begin{equation}\label{Spres}
(\mathscr{S}_\varphi A)_\pm=\mathscr{S}_\varphi(A_\pm), \quad \mathrm{res}\, (\mathscr{S}_\varphi A)=\mathrm{res}\, A, \quad (\mathscr{S}_\varphi A)^*=\mathscr{S}_{-\varphi}A^*.
\end{equation}

\section{The extended KP hierarchy and its bilinear equation}

We proceed to recall the definition of the extended KP hierarchy,
and then  capsule it into a bilinear equation of certain (adjoint)
Baker--Akhiezer functions.

\subsection{The extended KP hierarchy}
Let ${\cM}$ be an infinite-dimensional manifold with
coordinate
\[
\bm{a}=(a_1, a_2, a_3, \dots; f, b_1, b_2, b_3, \dots ).
\]
We consider the following graded
algebra of formal differential polynomials:
\begin{equation}\label{tA}
{\cA}=C^\infty(S^1\to{\cM})[[\p^r(\bm{a})\mid r\ge1]],
\end{equation}
in which the derivation is $\p=\od/\od x$ with $x$ being the
coordinate of the loop $S^1$, and
\[
\deg \bm{a}=0, \quad \deg \p^r(\bm{a})=r.
\]
Over the graded differential algebra $\cA$, it is defined the algebras
$\cD^-$ and  $\cD^+$ of pseudo-differential operators of the first
type and of the second type, respectively.

Let us consider two pseudo-differential operators as follows:
\begin{align} \label{Phi}
&\Phi=1+\sum_{i\ge 1}a_i \p^{-i}\in\cD^-,\\
&\hat{\Phi}=e^f \left(1+\sum_{i\ge 1}b_i \p^{i}\right)\in\cD^+. \label{Phih}
\end{align}
These two operators have inverses of the form
\begin{align} \label{Phiinv}
&\Phi^{-1}=1+\sum_{i\ge 1}\tilde{a}_i \p^{-i}\in\cD^-,\\
&\hat{\Phi}^{-1}=\left(1 +\sum_{i\ge 1}\tilde{b}_i
\p^{i}\right)e^{-f} \in\cD^+. \label{Phihinv}
\end{align}
In fact, by expanding $\Phi^{-1}\Phi=1$ and $\hat{\Phi}^{-1}\hat{\Phi}=1$, one sees that the coefficients $\tilde{a}_i$ and $\tilde{b}_i$ are represented as
\[
\tilde{a}_i=-a_i+g_i(a_1,a_2,\dots,a_{i-1}), \quad \tilde{b}_i=-b_i+h_i(b_1,b_2, b_3, \dots)
\]
with $g_i, h_i\in\cA_{\ge1}$.
For instance, one has $\tilde{a}_1=-a_1$, and that $\tilde{b}_1$ is given recursively by
\[
\left.\tilde{b}_1\right|_{\cA_0}=-b_1, \quad
\left.\tilde{b}_1\right|_{\cA_j}=- \sum_{r=1}^{j}b_r\left(
\left.\tilde{b}_1\right|_{\cA_{j-r}}\right)^{(r)} ~~ \hbox{ for } ~~
j\ge1.
\]
Here $\left.\tilde{b}_1\right|_{\cA_j}$ means the degree-$j$ component of $\tilde{b}_1$.

Now let us introduce two pseudo-differential operators as follows:
\begin{equation} \label{PPh}
P=\Phi \p\Phi^{-1}\in\cD^-,\quad \hat{P}=\hat{\Phi}
\p^{-1}\hat{\Phi}^{-1}\in\cD^+.
\end{equation}
\begin{prp}
The operators $P$ and $\hat{P}$ given above can be represented in
the form:
\begin{align}\label{PPh2}
P=\p+\sum_{i\ge1}v_i\p^{-i}, \quad
\hat{P}=(\p-f')^{-1}\rho+\sum_{i\ge0}\hat{v}_i\p^i,
\end{align}
where $v_{i+1}, \hat{v}_i\in \cA_{\ge1}$ for $i\in\Z_{\ge0}$, and
\begin{equation}\label{rho}
\rho=e^{f}\left(\hat\Phi^{-1}\right)^*(1).
\end{equation}
\end{prp}
\begin{prf}
The representation of $P$ is well known, so let us verify the case
of $\hat{P}$. To this end, firstly let us check that $\hat{P}$ given
in \eqref{PPh} takes the form
\begin{equation}\label{}
\hat{P}=(\p-f')^{-1}\left( \sum_{i\ge0}\tilde{v}_i(\p-f')^i\right), \quad
\tilde{v}_i-\delta_{i 0}\in \cA_{\ge1}.
\end{equation}
We substitute this expansion and \eqref{Phih} into an equivalent
version of $\hat{\Phi}\p^{-1}\hat{\Phi}^{-1}=\hat{P}$, say,
\[
e^{-f}(\p-f')\hat{\Phi}\p^{-1}=e^{-f}(\p-f')\hat{P}\hat{\Phi}.
\]
Note $e^{-f}(\p-f')e^{f}=\p$, then we have
\begin{align} \label{eqvtil}
1+b_1'+\sum_{i\ge1}(b_i+b_{i+1}')\p^{i}=\left(
\sum_{i\ge0}\tilde{v}_i\p^i\right)\left(1+ \sum_{i\ge1}b_i\p^i\right),
\end{align}
in which the coefficients of $\p^i$ lead to
\begin{align}\label{}
i=0 &: \quad 1+b_1'=\tilde{v}_0, \label{vtil-1} \\
i\ge1 &: \quad b_i+b_{i+1}'=\tilde{v}_i+\sum_{r=1}^i\sum_{s\ge 0}
\binom{i-r+s}{s}\tilde{v}_{i-r+s}b_{r}^{(s)}. \label{vtil}
\end{align}
The equations \eqref{vtil} yield
\[
b_i=\left.\tilde{v}_i\right|_{\cA_0}+\sum_{r=1}^i \left.\tilde{v}_{i-r}\right|_{\cA_0} b_r, \quad i\ge1,
\]
which together with \eqref{vtil-1} gives
\[
\left.\tilde{v}_i\right|_{\cA_0}=\delta_{i 0}, \quad i\ge0.
\]
The equations \eqref{vtil} also yield
\[
b_{i+1}'=\left.\tilde{v}_i\right|_{\cA_1}+\sum_{r=1}^i \left.\tilde{v}_{i-r}\right|_{\cA_1} b_r, \quad i\ge1,
\]
which together with \eqref{vtil-1} determine $\left.\tilde{v}_i\right|_{\cA_1}$ recursively. In fact, we can obtain a generating function for
$\left.\tilde{v}_i\right|_{\cA_1}$ of a parameter $z$ as
\[
\sum_{i\ge0} \left.\tilde{v}_i\right|_{\cA_1} z^{i+1}=\p \log\left(1+\sum_{i\ge 1}b_i z^i\right).
\]
For $j\ge2$, it follows from \eqref{vtil} that
\[
\left.\tilde{v}_i\right|_{\cA_j}+\sum_{r=1}^i\sum_{s=0}^{j-1} \binom{i-r+s}{s}\left.\tilde{v}_{i-r+s}\right|_{\cA_{j-s}}\cdot b_{r}^{(s)}=0, \quad i\ge1,
\]
hence $\left.\tilde{v}_i\right|_{\cA_j}$ are determined recursively.
By using \eqref{Svp}, we derive that the operator $\hat P$ takes the form \eqref{PPh2}, in which
\begin{equation}\label{}
\rho=\mathrm{res}\, \left(\hat{\Phi}\p^{-1}\hat{\Phi}^{-1}\right)=\mathrm{res}\, \left(e^f \p^{-1}\hat{\Phi}^{-1}\right)=e^f \left(\hat\Phi^{-1}\right)^*(1).
\end{equation}
Therefore the proposition is proved.
\end{prf}

\begin{rmk}
From \eqref{rho} and \eqref{Phihinv} it follows that
\begin{equation}\label{}
\rho=e^f \left(\hat\Phi^{-1}\right)^*(1)=1+\sum_{i\ge1}(-1)^i
\tilde{b}_i^{(i)},
\end{equation}
which implies $\rho-1\in\cA_{\ge1}$.
One also sees that, the coefficients of $P$ and $\hat{P}$ have different degrees from that in \cite{WZ}, since now the graded differential algebra $\cA$ is chosen differently.
\end{rmk}

With the help of the operators $P$ and $\hat{P}$, let us define a class of evolutionary equations on $\cM$: for $k\in\Z_{>0}$,
\begin{align}
&\frac{\pd \Phi}{\pd t_k}=- (P^k)_-\Phi, \quad
\frac{\pd \hat{\Phi}}{\pd t_k}=\bigl((P^k)_+ -\delta_{k1} \hat{P}^{-1}\bigr)\hat{\Phi}, \label{ppt1}\\
&\frac{\pd \Phi}{\pd \hat{t}_k}=- (\hat{P}^k)_-\Phi, \quad \frac{\pd
\hat{\Phi}}{\pd \hat{t}_k}=(\hat{P}^k)_+\hat{\Phi}, \label{ppt2}
\end{align}
Here we note that the right hand sides make sense since the operators $(P^k)_+, (\hat{P}^k)_-\in\cD^-\cap\cD^+$, and we assume that the flows $\p/\p t_k$ and $\p/\p\hat{t}_k$ commutate with $\p/\p x$. In particular, it can be seen $\p/\p t_1=\p/\p x$, so in what follows we will just take $t_1=x$.

\begin{prp}\label{thm-PPht}
The flows \eqref{ppt1}, \eqref{ppt2} satisfy, for $k\in\Z_{>0}$,
\begin{align}\label{PPht}
& \frac{\pd P}{\pd t_k}=\left[(P^k)_+, P\right], \quad \frac{\pd \hat{P}}{\pd
t_k}=\left[(P^k)_+, \hat{P}\right], \\
\label{PPhth} & \frac{\pd P}{\pd \hat{t}_k}=\left[-(\hat{P}^k)_-, P\right],
\quad \frac{\pd \hat{P}}{\pd \hat{t}_k}=\left[-(\hat{P}^k)_-, \hat{P}\right].
\end{align}
Moreover, these flows commute with each other.
\end{prp}

\begin{prf}
The proposition can be verified case by case. For instance, we have
\begin{align*}
\frac{\p\hat{P}}{\p t_k}= &\left[\frac{\p\hat\Phi}{\p t_k}{\hat\Phi}^{-1},{\hat P} \right]
= \left[(P^k)_+ -\delta_{k1} \hat{P}^{-1},{\hat P} \right]=\left[(P^k)_+ ,{\hat P} \right], \\
\left[\frac{\p}{\p t_k},\frac{\p}{\p{\hat t}_l} \right]{\hat\Phi}=& \frac{\p}{\p t_k}\left( (\hat{P}^l)_+\hat{\Phi}\right)-\frac{\p}{\p {\hat t}_l}\left( \bigl((P^k)_+ -\delta_{k1} \hat{P}^{-1}\bigr)\hat{\Phi} \right) \\
=& \left[(\hat{P}^l)_+, (P^k)_+ -\delta_{k1} \hat{P}^{-1} \right]\hat{\Phi}+ \left[(P^k)_+, \hat{P}^l\right]_+\hat{\Phi} \\
&- \left(\left[- (\hat{P}^l)_-, P^k\right]_+-\delta_{k1}\left[- (\hat{P}^l)_-, \hat{P}^{-1} \right] \right)\hat{\Phi} \\
=& \left[(\hat{P}^l)_+, (P^k)_+  \right]\hat{\Phi}-\left[ \hat{P}^l , (P^k)_+  \right]_+\hat{\Phi} \\
&+\left[(\hat{P}^l)_-, (P^k)_+  \right]_+\hat{\Phi} -\delta_{k1}\left[ \hat{P}^l , \hat{P}^{-1}  \right]\hat{\Phi} \\
=&0.
\end{align*}
The other cases are almost the same. So we complete the proof.
\end{prf}

The system of equations \eqref{PPht}, \eqref{PPhth} was studied in
\cite{WZ} by Zhou and one of the authors (cf. \cite{SB}), with the
set of unknown functions as
\[
\left\{ v_{i+1}, \hat{v}_i \mid i\in\Z_{\ge0}\right\}\cup\{\rho, \varphi=f'\}.
\]
This system is called the extended KP hierarchy for the reason that the flows $\p P/\p t_k$ in \eqref{PPht} with $k\in\Z_{>0}$ compose the well-known KP hierarchy. By virtue of the above proposition, we will also call the system of equations \eqref{ppt1}, \eqref{ppt2} the extended KP hierarchy.
\begin{exa}
From the extended KP hierarchy one can write down some equations explicitly as follows:
\begin{align}
\frac{\p^2 v_1}{\p {t_2}^2}=&\left(\frac{4}{3}\frac{\p v_1}{\p t_3}-4v_1 v_1' -\frac{1}{3}v_1''' \right)', \quad
\frac{\p f}{\p t_2}=2 v_1+(f')^2+f'', \quad
\frac{\p \rho}{\p t_2}= \left(2\rho f'-\rho'\right)'. \label{ekpt2}
\end{align}
\end{exa}

\begin{exa}
By using \eqref{Phih} and \eqref{ppt1} we have, for $k\in\Z_{>0}$,
\begin{align}
\frac{\p f}{\p t_k}=& e^{-f}\frac{\p\hat\Phi}{\p t_k}(1)
=e^{-f}\bigl((P^k)_+ -\delta_{k1} \hat{P}^{-1}\bigr)\hat{\Phi}(1)
= e^{-f}(P^k)_+  e^f (1)  \nonumber \\
=&\mathrm{res}\, \left(e^{-f}P^k   e^f\p^{-1}\right) =\mathrm{res}\, \mathscr{S}_{-f'}\left(P^k  e^f\p^{-1}e^{-f}\right)
\nonumber \\
=&\mathrm{res}\, \left(P^k  e^f\p^{-1}e^{-f}\right)= \mathrm{res}\, \left( P^k (\p-f')^{-1}\right). \label{ft}
\end{align}
Here in the fifth equality we have used \eqref{Spres}. With the same method, we obtain
\begin{align}
\frac{\p f}{\p\hat{t}_k}= \mathrm{res}\, \left( \hat{P}^k (\p-f')^{-1}\right), \quad k\in\Z_{>0}. \label{ft}
\end{align}
Moreover, for $\dot{t}_k=t_k$ or $\hat{t}_k$ (correspondingly, $\dot{P}=P$ or $\hat{P}$), we have
\begin{align}
\frac{\p\rho}{\p\dot{t}_k}=&\sum_{i\ge1}(-1)^{i-1}\p^i\left(\rho\,\mathrm{res}\,\,
\dot{P}^k(\p-f')^{-i-1}\right). \label{rhot}
\end{align}
By letting $\varphi=f'$, we recover the equations (2.23) and (2.24) in \cite{WZ}.
\end{exa}

\subsection{Baker--Akhiezer functions and the bilinear equation}

With $z$ being a nonzero parameter, we assign $\p^i\left( e^{x
z}\right)=z^i e^{x z}$ for any $i\in\Z$, and more generally,
\[
\p^{i}\left(g e^{x z}\right)=(\p^{i}g)\left( e^{x z}\right), \quad g\in\cA, ~~ i\in\Z.
\]
Namely, it is the usual action of a differential operator on a
function whenever $i\ge0$, and the integral constants are fixed in a
special way whenever $i<0$. In order to simplify notations, for any
$A\in\cD^\pm$ and exponential functions of the the form $e^{\pm x
z}$, we will just write $A e^{\pm x z}$ instead of $A\left( e^{\pm x
z}\right)$.

Denote $\bm{t}=(t_1=x,t_2,t_3,\dots)$  and
$\hat{\bm{t}}=(\hat{t}_1,\hat{t}_2,\hat{t}_3,\dots)$, and let $\xi$ be given by
\[
\xi(\bm{t}; z)=\sum_{k\in\Z_{>0}}
t_k z^k.
\]
Given a solution of the extended KP hierarchy \eqref{ppt1}, \eqref{ppt2}, let us introduce two
Baker--Akhiezer functions:
\begin{align}\label{wavef}
\psi(\bm{t}, \hat{\bm{t}}; z)=\Phi e^{\xi(\bm{t};z)}, \quad
\hat{\psi}(\bm{t}, \hat{\bm{t}}; z)=\hat{\Phi}  e^{x
z-\xi(\hat{\bm{t}};z^{-1})},
\end{align}
and two adjoint Baker--Akhiezer functions:
\begin{align}\label{baker}
\psi^\dag(\bm{t}, \hat{\bm{t}}; z)=\left(\Phi^{-1}\right)^*  e^{-\xi(\bm{t};z)} , \quad
\hat{\psi}^\dag(\bm{t}, \hat{\bm{t}}; z)=\left(\hat{\Phi}^{-1}\right)^*  e^{-x
z+\xi(\hat{\bm{t}};z^{-1})}.
\end{align}
When there is no confusion, we will just write $\eta(z)=\eta(\bm{t}, \hat{\bm{t}}; z)$   with $\eta\in\{\psi, \hat{\psi}, \psi^\dag, \hat{\psi}^\dag\}$. Based on \eqref{ppt1} and \eqref{ppt2},
it is straightforward to verify the the following Lemma.
\begin{lem}\label{thm-bakert}
The (adjoint) Baker--Akhiezer functions given above satisfy:
\begin{align}\label{}
\mathrm{(i)} \qquad & P\psi(z)=z\psi(z), \quad P^*\psi^\dag(z)=z\psi^\dag(z),  \\
& \hat{P}{\hat \psi}(z)=z^{-1}{\hat \psi}(z), \quad {\hat P}^*{\hat \psi}^\dag(z)=z^{-1}{\hat\psi}^\dag(z); \\
\mathrm{(ii)} \qquad
& \frac{\p\dot\psi(z)}{\p t_k}=\left(P^k\right)_+{\dot\psi(z)}, \quad \frac{\p\dot\psi(z)}{\p \hat{t}_k}=-\left(\hat{P}^k\right)_-\dot\psi(z),  \\
& \frac{\p\dot\psi^\dag(z)}{\p t_k}=-\left(P^k\right)_+^*{\dot\psi^\dag(z)}, \quad \frac{\p\dot\psi^\dag(z)}{\p \hat{t}_k}=\left(\hat{P}^k\right)_-^*\dot\psi^\dag(z)
\end{align}
where $\dot\psi\in\left\{\psi, \, \hat{\psi}\right\}$ and  $\dot\psi^\dag\in\left\{\psi^\dag, \, \hat{\psi}^\dag\right\}$.
\end{lem}

For any formal series $\sum_{i\in\Z}g_i z^i$, its residue is defined by
\[
\mathrm{res}\,_{\!z} \sum_{i\in\Z}g_i z^i=g_{-1}.
\]
The following lemma is useful.
\begin{lem}[see, for example, \cite{DKJM-KPBKP}] \label{thm-res}
For any pseudo-differential operators $Q, R\in\cD^\pm$, the following equality holds true whenever both sides make sense:
\begin{equation}\label{res}
\mathrm{res}\,_{\!z}\left(Q e^{z x}\cdot R^* e^{-z x}\right)=\mathrm{res}\, (Q R).
\end{equation}
\end{lem}

\begin{thm}\label{thm-bl}
The (adjoint) Baker--Akhiezer functions of the extended KP hierarchy satisfy
the following bilinear equation
\begin{equation}\label{bl}
\mathrm{res}\,_{\!z}\left(\psi(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)\psi^\dag(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}', \hbt'; z)\right)=\mathrm{res}\,_{\!z}\left(\hat{\psi}(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)\hat{\psi}^\dag(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}', \hbt'; z)\right),
\end{equation}
for arbitrary time variables $(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt)$ and $(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}', \hbt')$. Conversely, suppose that four  functions of the form
\begin{align}\label{wavef2}
&\psi(\bm{t}, \hat{\bm{t}}; z)=\left(1+\sum_{i\ge1}a_i(\bm{t}, \hat{\bm{t}})z^{-i}\right) e^{\xi(\bm{t};z)},  \\
&\hat{\psi}(\bm{t}, \hat{\bm{t}}; z)=e^{f(\bm{t}, \hat{\bm{t}}) }\left(1+\sum_{i\ge1}b_i(\bm{t}, \hat{\bm{t}})z^{i}\right) e^{x
z-\xi(\hat{\bm{t}};z^{-1})},  \\
&\psi^\dag(\bm{t}, \hat{\bm{t}}; z)=\left(1+\sum_{i\ge1} {a}_i^\dag(\bm{t}, \hat{\bm{t}})z^{-i}\right)   e^{-\xi(\bm{t};z)}, \\
&\hat{\psi}^\dag(\bm{t}, \hat{\bm{t}}; z)=e^{-f(\bm{t}, \hat{\bm{t}}) }\left(1+\sum_{i\ge1}{b}_i^\dag(\bm{t}, \hat{\bm{t}})z^{i}\right)  e^{-x
z+\xi(\hat{\bm{t}};z^{-1})} \label{wavef2a}
\end{align}
satisfy the bilinear equation \eqref{bl}, then they are the  Baker--Akhiezer functions and the adjoint  Baker--Akhiezer functions of the extended KP hierarchy.
\end{thm}
\begin{prf}
As a preparation, we introduce the set of indices as
\[
I=\left\{  (m_1,m_2,m_3,\dots )\mid m_i\in\Z_{\ge0} \hbox{ such that } m_i=0 \hbox{ for } i\gg 0\right\}.
\]
For $\bm{m}=(m_1,m_2,m_3,\dots )\in I$, denote
\[
{\p_\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}}^{\bm m}=\prod_{k\ge1}\left(\frac{\p}{\p {t_k} }\right)^{m_k}, \quad {\p_{\hbt}}^{\bm m}=\prod_{k\ge1}\left(\frac{\p}{\p \hat{t}_k }\right)^{m_k}.
\]
In order to show the equality \eqref{bl}, we only need to check that
the Baker--Akhiezer functions and the adjoint Baker--Akhiezer functions satisfy
\begin{equation}\label{bl-deriv}
\mathrm{res}\,_{\!z}\left({\p_\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}}^{\bm m}{\p_{\hbt}}^{\bm n}\psi(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)\cdot \psi^\dag(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)\right)=\mathrm{res}\,_{\!z}\left({\p_\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}}^{\bm m}{\p_{\hbt}}^{\bm n}\hat{\psi}(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)\cdot \hat{\psi}^\dag(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)\right)
\end{equation}
for any indices ${\bm m}, {\bm n}\in I$. In fact, given any ${\bm m}, {\bm n}\in I$, according to Lemma~\ref{thm-bakert} and Proposition~\ref{thm-PPht} there is a pseudo-differential operator $A^{\bm{m}, \bm{n}}\in \cD^-\cap\cD^+$ such that the following two equalities hold simultaneously:
\[
{\p_\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}}^{\bm m}{\p_{\hbt}}^{\bm n}\psi(z)=A^{\bm{m}, \bm{n}}\psi(z), \quad {\p_\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}}^{\bm m}{\p_{\hbt}}^{\bm n}\hat\psi(z)=A^{\bm{m}, \bm{n}}\hat\psi(z).
\]
Then, by using \eqref{baker} and Lemma~\ref{thm-res}, the equality \eqref{bl-deriv} is recast to
\[
\mathrm{res}\,\left( A^{\bm{m}, \bm{n}}\Phi \Phi^{-1}\right)=\mathrm{res}\,\left( A^{\bm{m}, \bm{n}}\hat\Phi {\hat\Phi}^{-1}\right),
\]
which is clearly valid. Hence the equality \eqref{bl-deriv} holds
true, and the first assertion is verified.

For the second assertion, one sees that there are uniquely
pseudo-differential operators $\Phi$, $\hat{\Phi}$, $\Psi$ and
$\hat{\Psi}$ such that the functions \eqref{wavef2}--\eqref{wavef2a}
are represented as
\begin{align*}
\psi(\bm{t}, \hat{\bm{t}}; z)=\Phi e^{\xi(\bm{t};z)},& \quad
\hat{\psi}(\bm{t}, \hat{\bm{t}}; z)=\hat{\Phi}  e^{x
z-\xi(\hat{\bm{t}};z^{-1})},
\\
\psi^\dag(\bm{t}, \hat{\bm{t}}; z)=\Psi^*  e^{-\xi(\bm{t};z)},& \quad
\hat{\psi}^\dag(\bm{t}, \hat{\bm{t}}; z)=\hat{\Psi}^*  e^{-x
z+\xi(\hat{\bm{t}};z^{-1})}.
\end{align*}
Moreover, the operators $\Phi$ and $\hat\Phi$ take the form
\eqref{Phi} and \eqref{Phih} respectively, while $\Psi$ and
$\hat\Psi$ take the form \eqref{Phiinv} and \eqref{Phihinv}
respectively. The bilinear equation \eqref{bl} leads to the
following facts.
\begin{itemize}
\item[(i)] For $i\in\Z$, we have (note that when $i<0$ the integral constants on both sides are fixed in the same way)
\[
\mathrm{res}\,_{\!z}\left(\p^i\psi(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)\psi^\dag(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}', \hbt'; z)\right)=\mathrm{res}\,_{\!z}\left(\p^i\hat{\psi}(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt; z)\hat{\psi}^\dag(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}', \hbt'; z)\right).
\]
Let $(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}', \hbt')=(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt)$, and then with the help of \eqref{res} we obtain
\begin{align*}
\mathrm{res}\, \p^i\Phi\Psi=\mathrm{res}\, \p^i\hat{\Phi}\hat{\Psi}=0,& \quad i{\ge0}; \\
\mathrm{res}\, \p^i\hat{\Phi}\hat{\Psi}=\mathrm{res}\, \p^i\Phi\Psi=\delta_{i,-1},& \quad i<0.
\end{align*}
which implies $\Phi\Psi=\hat{\Phi}\hat{\Psi}=1$. So we derive
\begin{equation}\label{}
\Psi= \Phi^{-1} , \quad \hat{\Psi}= \hat{\Phi}^{-1}.
\end{equation}
\item[(ii)]
Denote
\[
X_k=\frac{\p\Phi}{\p t_k}\Phi^{-1}, \quad
\hat{X}_k=\frac{\p\hat{\Phi}}{\p t_k}\hat{\Phi}^{-1}.
\]
Clearly, one has $(X_k)_+=(\hat{X}_k)_-=0$, and that
\begin{align*}
&\frac{\p\psi(z)}{\p t_k}=\left(X_k\Phi +\Phi \p^k\right)e^{\xi(\bm{t};z)}=\left(X_k +\Phi \p^k\Phi^{-1}\right)\psi(z), \\
&\frac{\p\hat{\psi}(z)}{\p t_k}= \left(\hat{X}_k\hat{\Phi}+\delta_{k
1}\hat{\Phi}\p \right) e^{x z-\xi(\hat{\bm{t}};z^{-1})}=
\left(\hat{X}_k+\delta_{k 1}\hat{\Phi}\p\hat{\Phi}^{-1}
\right)\hat{\psi}(z).
\end{align*}
For any $i\in\Z$, we let $\p^i$ act on the derivative of \eqref{bl} with respect to $t_k$, and let $(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}', \hbt')=(\bm{t}}  \newcommand{\hbt}{\hat{\bm{t}}, \hbt)$, then by using \eqref{res} again we obtain
\begin{align*}
\mathrm{res}\, \p^i\left(X_k +\Phi \p^k\Phi^{-1}\right)\Phi\Phi^{-1}=\mathrm{res}\, \p^i
\left(\hat{X}_k+\delta_{k 1}\hat{\Phi}\p\hat{\Phi}^{-1}
\right)\hat{\Phi}\hat{\Phi}^{-1}.
\end{align*}
Hence $X_k +\Phi \p^k\Phi^{-1}=\hat{X}_k+\delta_{k
1}\hat{\Phi}\p\hat{\Phi}^{-1} $, and we arrive at
\begin{equation}\label{}
X_k=-\left( \Phi \p^k\Phi^{-1}\right)_-, \quad \hat{X}_k=\left( \Phi
\p^k\Phi^{-1}\right)_+-\delta_{k 1}\hat{\Phi}\p\hat{\Phi}^{-1} .
\end{equation}
Similarly, we can derive
\begin{equation}\label{}
\frac{\p\Phi}{\p\hat{t}_k}\Phi^{-1}=-\left( \hat\Phi \p^{-k}\hat\Phi^{-1}\right)_-, \quad \frac{\p\hat{\Phi}}{\p \hat{t}_k}\hat{\Phi}^{-1}=\left( \hat\Phi \p^{-k}\hat\Phi^{-1}\right)_+.
\end{equation}
\end{itemize}
Taking (i) and (ii) together we achieve the second assertion. The
theorem is proved.
\end{prf}

\section{Additional symmetries versus Virasoro symmetries}

In this section, we want to construct a class of additional
symmetries for the extended KP hierarchy,
following the approach of \cite{OS, Wu-virD}, and then study the
Virasoro symmetries for the constrained KP hierarchies.

\subsection{Additional symmetries for the extended KP hierarchy}
Suppose that the operators $\Phi$ and $\hat{\Phi}$ solve the hierarchy
\eqref{ppt1}, \eqref{ppt2}. Let us introduce two Orlov--Schulman
operators as follows:
\begin{equation}\label{MMh}
M=\Phi\Xi\Phi^{-1}, \quad
\hat{M}=\hat{\Phi}\hat{\Xi}\hat{\Phi}^{-1},
\end{equation}
where
\[
\Xi=\sum_{k\in\Z_{>0}}k t_k  \p^{k-1}, \quad
\hat{\Xi}=x+\sum_{k\in\Z_{>0}} k\hat{t}_k \p^{-k-1}.
\]
Here we assume that all $t_k$ and $\hat{t}_k$ vanish except finitely
many of them, such that the  operators $M$ and $\hat{M}$ are well defined.

\begin{rmk}
There is another way to ensure the Orlov--Schulman operators $M$ and
$\hat{M}$ (even with infinitely many time variables) to be well
defined. Indeed, as what was done in \cite{Wu-virD}, one can extend
the graded algebra $\cA$ to include also  $\{t_k, \hat{t}_k\mid
k\in\Z_{>0}\}$ with $\deg t_k=\deg\hat{t}_k=k$, so that a new graded
algebra  $\tilde{\cA}$ is obtained. Then, the set $\cD^-$ of
pseudo-differential operators of the first type can be extended to
\[
\tilde{\cD}^-=\left\{ \sum_{i\in\Z}\sum_{j\ge \max\{0,i-k\}}a_{i,j} \p^i
\mid a_{i,j}\in\tilde{\cA}_j, k\in\Z \right\}.
\]
So, the operators $M$ and $\hat{M}$ are elements of $\tilde{\cD}^-$
and $\cD^+$ (with $\cA$ replaced by $\tilde{\cA}$) respectively.
\end{rmk}

\begin{lem}\label{thm-Mw}
The Orlov--Schulman operators $M$ and $\hat{M}$ satisfy:
\begin{align}\label{LMLMh}
[P, M]=1,& \quad [\hat{P}^{-1},\hat{M}]=1, \\
M \psi(z)=\frac{\p\psi(z)}{\p z},& \quad \hat{M} \hat{\psi}(z)=\frac{\p\hat{\psi}(z)}{\p z}, \label{Mpsi}  \\
\frac{\pd \dot{M}}{\pd t_k}=[(P^k)_+,\dot{M}], & \quad
 \frac{\pd \dot{M}}{\pd \hat{t}_k}=[-(\hat{P}^k)_-, \dot{M}], \label{Mt}
\end{align}
where  $\dot{M}=M$ or $\hat{M}$.
\end{lem}
\begin{prf}
Based on the definition of $M$ and $\hat{M}$ in \eqref{MMh}, the
first line \eqref{LMLMh} follows from \eqref{PPh}, the second line
\eqref{Mpsi} follows from \eqref{wavef}, while the third line
\eqref{Mt} follows from \eqref{ppt1} and \eqref{ppt2}. The lemma is
proved.
\end{prf}

For any pair of integers $(m,p)$ with $m\ge0$, let
\begin{align}\label{Aml}
& B_{m p}=M^m P^p, \quad
\hat{B}_{m p}=\hat{M}^m \hat{P}^{-p},
\end{align}
and we introduce the following evolutionary equations:
\begin{align}
&\frac{\pd \Phi}{\pd \beta_{m p}}=- (B_{m p})_-\Phi,
\quad \frac{\pd \hat{\Phi}}{\pd \beta_{m p}}=(B_{m p})_+\hat{\Phi}, \label{sml}\\
&\frac{\pd \Phi}{\pd \hat{\beta}_{m p}}=- (\hat{B}_{m p})_-\Phi,
\quad \frac{\pd \hat{\Phi}}{\pd \hat{\beta}_{m p}}=(\hat{B}_{m
p})_+\hat{\Phi}. \label{shml}
\end{align}
As before, such flows are assumed to commute with $\pd/\pd x$.

\begin{lem}\label{thm-MLs}
For any $m, m'\in\Z_{\ge0}$ and $p, p'\in\Z$, the following equalities hold:
\begin{align}
&\frac{\pd \psi(z)}{\pd \dot{\beta}_{m p}}=-(\dot{B}_{m
p})_-\psi(z), \quad \frac{\pd \hat{\psi}(z)}{\pd \dot{\beta}_{m
p}}=(\dot{B}_{m p})_+\hat{\psi}(z), \label{wsdot}
\\
&\frac{\pd P}{\pd \dot{\beta}_{m p}}=[-(\dot{B}_{m p})_-, P], \quad
\frac{\pd \hat{P}}{\pd \dot{\beta}_{m p}}=[(\dot{B}_{m p})_+,
\hat{P}], \label{Lsdot}
\\
&\frac{\pd M}{\pd \dot{\beta}_{m p}}=[-(\dot{B}_{m p})_-, M], \quad
\frac{\pd \hat{M}}{\pd \dot{\beta}_{m p}}=[(\dot{B}_{m p})_+,
\hat{M}], \label{Msdot}
\\
&\frac{\pd B_{m'p'}}{\pd \dot{\beta}_{m p}}=[-(\dot{B}_{m p})_-,
B_{m'p'}], \quad \frac{\pd \hat{B}_{m'p'}}{\pd \dot{\beta}_{m
p}}=[(\dot{B}_{m p})_+, \hat{B}_{m'p'}], \label{Asdot}
\end{align}
where $\dot{\beta}_{m p}=\beta_{m p}, \hat{\beta}_{m p}$ correspond
to $\dot{B}_{m p}=B_{m p}, \hat{B}_{m p}$ respectively.
\end{lem}
\begin{prf}
The equalities \eqref{wsdot}--\eqref{Msdot} follow from the
definition \eqref{sml}, \eqref{shml}. Subsequently,  the equalities \eqref{Asdot} follow from
\eqref{Lsdot} and \eqref{Msdot}. The lemma is
proved.
\end{prf}

\begin{thm}\label{thm-st}
The flows defined by \eqref{sml}, \eqref{shml} commute with those in
\eqref{ppt1}, \eqref{ppt2} that compose the extended KP
hierarchy. More exactly, for any $\dot{\beta}_{m p}=\beta_{m p},
\hat{\beta}_{m p}$ and $\bar{t}_k=t_k, \hat{t}_k$ it holds that
\begin{equation}\label{st}
\left[\frac{\pd}{\pd \dot{\beta}_{m p}}, \frac{\pd}{\pd
\bar{t}_k}\right]\tilde\Phi=0, \quad m\in\Z_{\ge0}, ~~ p\in\Z, ~~ k\in\Z_{>0},
\end{equation}
where $\tilde\Phi=\Phi$ or $\hat\Phi$.
\end{thm}
\begin{prf}
Firstly, from \eqref{PPht}, \eqref{PPhth} and \eqref{Mt} it follows that
\[
\frac{\pd \dot{B}_{m p}}{\pd t_k}=[(P^k)_+,\dot{B}_{m p}],  \quad
 \frac{\pd \dot{B}_{m p}}{\pd \hat{t}_k}=[-(\hat{P}^k)_-, \dot{B}_{m p}],
\]
with $ \dot{B}_{m p}= {B}_{m p}$, $ \hat{B}_{m p}$. Then the
proposition is checked case by case with the help of
Lemmas~\ref{thm-Mw} and \ref{thm-MLs}. For instance,
\begin{align}\label{}
\left[\frac{\pd}{\pd \hat{\beta}_{m p}}, \frac{\pd}{\pd
t_k}\right]\hat{\Phi}=&\frac{\pd}{\pd \hat{\beta}_{m p}} \left(((P^k)_+-\delta_{k1}\hat{P}^{-1} )\hat{\Phi} \right)-\frac{\pd}{\pd
t_k} \left( (\hat{B}_{m p})_+\hat{\Phi} \right) \nonumber\\
=& [ (P^k)_+-\delta_{k1}\hat{P}^{-1},(\hat{B}_{m p})_+]\hat{\Phi}
 \nonumber\\
 &+\left([-(\hat{B}_{m p})_-, P^k]_+ - \delta_{k1}[(\hat{B}_{m p})_+,
\hat{P}^{-1}]\right)\hat{\Phi}   -[(P^k)_+,\hat{B}_{m p}]_+\hat{\Phi} \nonumber \\
=&\left([ (P^k)_+,(\hat{B}_{m p})_+]+[ (P^k)_+,(\hat{B}_{m p})_-]_+
-[ (P^k)_+,\hat{B}_{m p}]_+ \right)\hat{\Phi} =0. \nonumber
\end{align}
The other cases are similar. Thus the proposition is proved.
\end{prf}

Proposition~\ref{thm-st} means that the flows
\eqref{sml}, \eqref{shml} give a class of symmetries for the extended KP
hierarchy, which are called the \emph{additional
symmetries}.

Let us study the commutation relation between the additional
symmetries themselves. Observe that each commutator $[B_{m p}, B_{m'
p'}]$ is a polynomial in $M$ and $P^{\pm1}$, hence, by virtue of
\eqref{LMLMh}, there exist certain constants $c_{m p, m' p'}^{n q}$
such that
\begin{equation}\label{BB}
[B_{m p}, B_{m' p'}]=\sum_{n, q}c_{m p, m' p'}^{n q}B_{n q}.
\end{equation}
In fact, one has $c_{m p, m' p'}^{n q}=0$ whenever $n\ge m+m'$ or
$|q-(p+p')|>\max(m, m')$, which implies that all but finitely many
structure constants on the right hand side of \eqref{BB} vanish. For
instance, when $m+m'\le2$ one has
\begin{align*}
c_{0p,0p'}^{n q}=0, \quad &c_{0p, 1p'}^{n
q}=p\delta_{n0}\delta_{q,p+p'-1}, \quad
c_{1 p, 1 p'}^{n q}=(p-p')\delta_{n1}\delta_{q, p+p'-1}, \\
&c_{0 p, 2 p'}^{n q}=p(p-1)\delta_{n0}\delta_{q,p+p'-2}+2p \delta_{n1}\delta_{q,
p+p'-1}.
\end{align*}
By virtue of \eqref{LMLMh}, it holds for the same structure
constants that
\begin{equation}\label{BhBh}
[\hat{B}_{m p}, \hat{B}_{m' p'}]=\sum_{n,q}c_{m p, m' p'}^{n
q}\hat{B}_{n q}.
\end{equation}

\begin{prp}\label{thm-ssh}
For the extended KP hierarchy \eqref{ppt1}, \eqref{ppt2}, its additional symmetries defined by \eqref{sml}, \eqref{shml} satisfy:
\begin{align}
&\left[\frac{\pd}{\pd \beta_{m p}}, \frac{\pd}{\pd \beta_{m'
p'}}\right]\tilde\Phi=-\sum_{n,q}c_{m p, m' p'}^{n q}\frac{\pd\tilde\Phi}{\pd \beta_{n
q}},
\\
& \left[\frac{\pd}{\pd \hat{\beta}_{m p}}, \frac{\pd}{\pd
\hat{\beta}_{m'
p'}}\right]\tilde\Phi=\sum_{n,q}c_{m p, m' p'}^{n q}\frac{\pd\tilde\Phi}{\pd \hat{\beta}_{n q}},\\
 &\left[\frac{\pd}{\pd \beta_{m p}}, \frac{\pd}{\pd \hat{\beta}_{m'
p'}}\right]\tilde\Phi=0,
\end{align}
where $\tilde\Phi=\Phi$ or $\hat\Phi$.
\end{prp}
\begin{prf}
The conclusion can be checked case by case with the help of
Lemma~\ref{thm-MLs}. For instance,
\begin{align}\label{}
&\left[\frac{\pd}{\pd \beta_{m p}}, \frac{\pd}{\pd \beta_{m' p'}}\right]\hat{\Phi} \nonumber\\
=& [ (B_{m'p'})_+, (B_{m p})_+]\hat{\Phi} + [-(B_{m p})_-,
B_{m'p'}]_+\hat{\Phi} - [-(B_{m'p'})_-, B_{m p}]_+\hat{\Phi} \nonumber \\
=& -[(B_{m p})_+, (B_{m'p'})_+]\hat{\Phi} - [(B_{m p})_-,
(B_{m'p'})_+]_+\hat{\Phi} - [B_{m p}, (B_{m'p'})_-]_+\hat{\Phi} \nonumber
\\
=&-[B_{m p}, B_{m'p'}]_+\hat{\Phi}
=-\sum_{n,q}c_{m p, m' p'}^{n q}(B_{n q})_+\hat{\Phi} \nonumber\\
=&-\sum_{n,q}c_{m p, m' p'}^{n q}\frac{\pd\hat{\Phi}}{\pd \beta_{n
q}},
\\
&\left[\frac{\pd}{\pd \hat{\beta}_{m p}}, \frac{\pd}{\pd \hat{\beta}_{m' p'}}\right]\hat{\Phi} \nonumber\\
=& [ (\hat{B}_{m'p'})_+, (\hat{B}_{m p})_+]\hat{\Phi} + [(\hat{B}_{m
p})_+,
\hat{B}_{m'p'}]_+\hat{\Phi} - [(\hat{B}_{m'p'})_+, \hat{B}_{m p}]_+\hat{\Phi} \nonumber \\
=&  [(\hat{B}_{m p})_+, (\hat{B}_{m'p'})_-]_+\hat{\Phi} +
[\hat{B}_{m p}, (\hat{B}_{m'p'})_+]_+\hat{\Phi} \nonumber
\\
=&[\hat{B}_{m p}, \hat{B}_{m'p'}]_+\hat{\Phi} = \sum_{n,q}c_{m p, m'
p'}^{n q}(\hat{B}_{n q})_+\hat{\Phi} \nonumber
\\
=&\sum_{n,q}c_{m p, m' p'}^{n q}\frac{\pd\hat{\Phi}}{\pd
\hat{\beta}_{n
q}}, \\
&\left[\frac{\pd}{\pd \beta_{m p}}, \frac{\pd}{\pd\hat{\beta}_{m' p'}}\right]\hat{\Phi} \nonumber\\
=& [ (\hat{B}_{m'p'})_+, (B_{m p})_+]\hat{\Phi} + [(B_{m p})_+,
\hat{B}_{m'p'}]_+\hat{\Phi} - [-(\hat{B}_{m'p'})_-, B_{m p}]_+\hat{\Phi} \nonumber \\
=& \left([(B_{m p})_+, (\hat{B}_{m'p'})_-]_+ + [(\hat{B}_{m'p'})_-,
(B_{m p})_+]_+\right)\hat{\Phi} =0.
\end{align}
The other cases are checked in the same way. Thus the proposition is
proved.
\end{prf}

This proposition means that the additional symmetries \eqref{sml},
\eqref{shml} for the extended KP hierarchy generate a
$W_{1+\infty}\times W_{1+\infty}$ algebra.

\subsection{Virasoro symmetries for the constrained KP hierarchies}

Given an arbitrary positive integer $n$, let us consider  the extended KP hierarchy \eqref{ppt1}, \eqref{ppt2} imposed with the following
constraint
\begin{equation}\label{PnPh}
 \Phi\p^n\Phi^{-1}=\hat{\Phi}\p^{-1}\Phi^{-1}.
\end{equation}
Under this constraint, one has $P^n=\hat{P}$ and hence $\p/\p t_{n k}=\p/\p \hat{t}_k$ for
$k\ge1$. Consequently, the extended KP hierarchy is reduced to
\begin{equation}\label{Lt}
\frac{\p L}{\p t_k}=[(P^k)_+,L], \quad k=1,2,3,\dots,
\end{equation}
where $L:=P^n=\hat{P}$ takes the form
\begin{equation}\label{L}
L=\p^n+u_{1}\p^{n-2}+u_2\p^{n-3}+\dots+u_{n-2}\p+u_{n-1}+(\p-f')^{-1}\rho.
\end{equation}
The system of equations \eqref{Lt} is called the constrained KP hierarchy,
denoted by $\hbox{cKP}_{n,1}$ (see \cite{ANP, BX, Ch, Di}). In fact, if
we write $v=e^f$ and $w=\rho e^{-f}$, then
\begin{equation}\label{}
L_-=v\p^{-1}w
\end{equation}
and it is yielded by the equations \eqref{Lt} that
\begin{equation}\label{}
\frac{\p v}{\p t_k}=\left(P^k\right)_+(v), \quad \frac{\p w}{\p t_k}=-\left(P^k\right)^*_+(w).
\end{equation}

\begin{exa}
The hierarchy
\eqref{Lt} is known as the nonlinear Shr\"{o}dinger hierarchy (see, e.g., \cite{Ch}) when $n=1$, and it is the Yajima--Oikawa hierarchy
\cite{YO} when
$n=2$. For instance, when $n=2$ the operator $L$ takes the form
\[
L=\p^2+u+(\p-f')^{-1}\rho=\p^2+u+v\p^{-1}w,
\]
then the first nontrivial equations
defined by \eqref{Lt} read (cf. \eqref{ekpt2})
\begin{equation}\label{rhophit3}
\frac{\p u}{\p t_2}=2 \rho', \quad \frac{\p\rho}{\p
t_2}=(2\rho f'-\rho')', \quad \frac{\p f}{\p t_2}=u+(f')^2+f'',
\end{equation}
or equivalently,
\begin{equation}\label{rhophit3}
\frac{\p u}{\p t_2}=2 (v w)', \quad \frac{\p v}{\p
t_2}=v''+u v, \quad \frac{\p w}{\p t_2}=-w''-u w,
\end{equation}
which is called the Yajima--Oikawa system.
\end{exa}

We proceed to construct a series of Virasoro symmetries for the
$\hbox{cKP}_{n,1}$ hierarchy \eqref{Lt} with the help of the extended KP hierarchy \eqref{ppt1}, \eqref{ppt2}.
To this end, let us introduce a class of operators as
\begin{equation}\label{Sp}
S_p=\frac{1}{n}M P^{n p+1}+\hat{M}\hat{P}^{p-1}
\quad p\in\Z,
\end{equation}
where
\[
M=\Phi\left(\sum_{k\in\Z_{>0}}k t_k\p^{k-1}\right)\Phi^{-1}, \quad  \hat{M}=\hat{\Phi}x\hat{\Phi}^{-1}.
\]
Here the operator $M$ takes the same form as in \eqref{MMh}, while for convenience $\hat{M}$ does not contain the time variables $\hat{t}_k$ as before. One observes that the operators $S_p$ belong to the space $\cD^- +\cD^+$, hence $(S_p)_-\in\cD^-$ and $(S_p)_+\in\cD^+$. The following evolutionary equations are well defined:
\begin{equation}\label{Phis}
\frac{\pd \Phi}{\pd s_p}=-(S_p)_-\Phi, \quad
\frac{\pd\hat{\Phi}}{\pd s_p}=(S_p)_+\hat{\Phi}, \qquad p\ge-1.
\end{equation}

\begin{prp}\label{thm-cKPvir}
The flows defined by \eqref{Phis} are consistent with the constraint \eqref{PnPh}, and they are reduced to
\begin{equation}\label{Ls}
\frac{\p L}{\p s_p}=\left[-(S_p)_-, L\right]=\left[(S_p)_+, L\right], \quad p\ge-1,
\end{equation}
Moreover, the reduced flows defined by  \eqref{Phis} and by \eqref{ppt1}  under the constraint \eqref{PnPh} satisfy:
\begin{align}\label{}
\mathrm{(i)} \qquad & \left[\frac{\p}{\p t_k}, \frac{\p}{\p s_p}\right]L=0,  \\
\mathrm{(ii)} \qquad
& \left[\frac{\p}{\p s_p}, \frac{\p}{\p s_q}\right]L=(q-p)\frac{\p L}{\p s_{p+q}}, \label{ss}
\end{align}
where $k\in\Z_{>0}$ and  $p, q\in\Z_{\ge-1}$.
\end{prp}

\begin{prf}
Since
\begin{equation}\label{LPhi}
L=\Phi\p^n\Phi^{-1}=\hat{\Phi}\p^{-1}\Phi^{-1},
\end{equation}
then the equations in \eqref{Phis} lead respectively to
\begin{equation}\label{Ls2}
\frac{\p L}{\p s_p}=\left[-(S_p)_-, L\right], \quad \frac{\p L}{\p s_p}=\left[(S_p)_+, L\right].
\end{equation}
When $p\ge-1$, it is straightforward to verify
\begin{equation}\label{SL}
[S_p, L]=\frac{1}{n}\left[M P^{n p+1}, P^n\right]+\left[\hat{M}\hat{P}^{p-1}, \hat{P}\right] =-P^{n(p+1)}+\hat{P}^{p+1}=-L^{p+1}+L^{p+1}=0,
\end{equation}
which implies that the equations in \eqref{Ls2} coincide with each other.
Thus the first assertion is obtained.

Let us show the second assertion. On the one hand, by using \eqref{ppt1} we have, for $p\ge-1$ and $k\ge1$,
\begin{align}
\frac{\p S_p}{\p t_k}=& \frac{1}{n}\left[ -(P^k)_-, M P^{n p+1} \right]+\frac{k}{n}P^{k+n p} \nonumber\\
& +  \left[ (P^k)_+-\delta_{k 1}\hat{P}^{-1}, \hat{M} \hat{P}^{p-1} \right] +\delta_{k 1}\hat{P}^{p-1}\nonumber\\
=&\frac{1}{n}\left[ -(P^k)_-+P^k, M P^{n p+1} \right]+  \left[ (P^k)_+, \hat{M} \hat{P}^{p-1} \right] \nonumber\\
=&\left[(P^k)_+, S_p\right],
\end{align}
On the other hand, by using \eqref{Phis} we have, for $p,q\ge-1$,
\begin{equation}\label{}
\frac{\p (M P^p)}{\p s_q}=\left[-(S_q)_-,M P^p\right], \quad \frac{\p (\hat{M} \hat{P}^p)}{\p s_q}=\left[(S_q)_+,\hat{M} \hat{P}^p\right].
\end{equation}
Then the first item is checked as
\begin{align*}
\left[\frac{\p}{\p t_k}, \frac{\p}{\p s_p}\right]L=& \frac{\p}{\p t_k}\left[(S_p)_+, L\right]- \frac{\p}{\p s_p}\left[(P^k)_+, L\right]  \\
=& \left[\left[(P^k)_+, S_p\right]_+, L\right]+ \left[ (S_p)_+, \left[(P^k)_+,L\right]\right] \\
&\quad - \left[\left[-(S_p)_-, P^k\right]_+, L\right]- \left[(P^k)_+, \left[ (S_p)_+,L\right]\right] \\
=& \left[\left[(P^k)_+, (S_p)_+\right], L\right]- \left[ \left[(P^k)_+,L\right],  (S_p)_+\right] - \left[(P^k)_+, \left[ (S_p)_+,L\right]\right]=0.
\end{align*}
For the second item, it is straightforward to calculate
\begin{align*}
\left[\frac{\p}{\p s_p}, \frac{\p}{\p s_q}\right]L=& \frac{\p}{\p s_p}\left[(S_q)_+, L\right]-
\frac{\p}{\p s_q}\left[(S_p)_+, L\right]  \\
=& \left[\left[-(S_p)_-, \frac{1}{n}M P^{n q+1}\right]_+ +\left[(S_p)_+, \hat{M}\hat{P}^{q-1}\right]_+, L\right] \\
&\quad + \left[ (S_q)_+, \left[(S_p)_+,L\right]\right]-(p\leftrightarrow q) \\
=& \Big[\left[-(S_p)_-, \frac{1}{n}M P^{n q+1}\right]_+ +\left[(S_p)_+, \hat{M}\hat{P}^{q-1}\right]_+ -\left[-(S_q)_-, \frac{1}{n}M P^{n p+1}\right]_+  \\
&\quad  -\left[(S_q)_+, \hat{M}\hat{P}^{p-1}\right]_+ + \left[ (S_q)_+,  (S_p)_+\right], L\Big] \\
=&\left[\frac{1}{n^2}X +\frac{1}{n}Y+Z, L\right]
\end{align*}
where, with \eqref{Sp} substituted,
\begin{align*}
X=& \left[-(M P^{n p+1})_-, M P^{n q+1}\right]_+ - \left[-(M P^{n q+1})_-, M P^{n p+1}\right]_+  \\
&\quad  + \left[(M P^{n q+1})_+, (M P^{n p+1})_+\right]  \\
=&  \left[M P^{n q+1} , M P^{n p+1} \right]_+ =n(q-p)(M P^{n (p+q)+1})_+,  \\
Y=& \left[-(\hat{M}\hat{P}^{p-1})_-, M P^{n q+1}\right]_+ + \left[(M P^{n p+1})_+,\hat{M}\hat{P}^{q-1}\right]_+  \\
&- \left[-(\hat{M}\hat{P}^{q-1})_-, M P^{n p+1}\right]_+ -\left[(M P^{n q+1})_+,\hat{M}\hat{P}^{p-1}\right]_+ \\
&+ \left[(M P^{n q+1})_+, (\hat{M}\hat{P}^{p-1})_+\right] + \left[(\hat{M}\hat{P}^{q-1})_+, (M P^{n p+1})_+\right]  \\
=& \left[(M P^{n q+1})_+, \hat{M}\hat{P}^{p-1}, \right]_+ - \left[(M P^{n q+1})_+,\hat{M}\hat{P}^{p-1}\right]_+  \\
 & \quad + \left[(M P^{n p+1})_+,(\hat{M}\hat{P}^{q-1})_+\right]_+    + \left[(\hat{M}\hat{P}^{q-1})_+, (M P^{n p+1})_+\right] =0, \\
Z=&\left[(\hat{M}\hat{P}^{p-1})_+, \hat{M}\hat{P}^{q-1}\right]_+ -\left[(\hat{M}\hat{P}^{q-1})_+, \hat{M}\hat{P}^{p-1}\right]_+ + \left[(\hat{M}\hat{P}^{q-1})_+, (\hat{M}\hat{P}^{p-1})_+\right] \\
=&\left[(\hat{M}\hat{P}^{p-1})_+, \hat{M}\hat{P}^{q-1}\right]_+ -\left[(\hat{M}\hat{P}^{q-1})_+, (\hat{M}\hat{P}^{p-1})_-\right]_+ \\
=&\left[\hat{M}\hat{P}^{p-1} , \hat{M}\hat{P}^{q-1}\right]_+=(q-p)(\hat{M}\hat{P}^{p+q-1})_+.
\end{align*}
So we obtain
\begin{align*}
\left[\frac{\p}{\p s_p}, \frac{\p}{\p s_q}\right]L=(q-p)\left[ (S_{p+q})_+, L\right]=(q-p)\frac{\p L}{\p s_{p+q}}.
\end{align*}
The proposition is proved.
\end{prf}

According to this proposition, the flows \eqref{Ls} give a series of Virasoro symmetries for the  $\hbox{cKP}_{n,1}$ hierarchy \eqref{Lt}. It is worthwhile to indicate that, here the condition $p\ge-1$ is necessary, otherwise \eqref{SL} may not vanish since $L$ has different inverse as an operator in $\cD^-$ or in $\cD^+$.

One observes that the flows $\pd/\pd s_p$ in \eqref{Phis} can be considered as reductions of the linear combinations
\[
\frac{1}{n}\frac{\pd}{\pd
\beta_{1,n p+1}}+\frac{\pd}{\pd \hat{\beta}_{1,-p+1}}
\]
of the additional symmetries \eqref{sml}, \eqref{shml} for the
extended KP hierarchy constrained by \eqref{LPhi}. This observation motivates us to consider whether the symmetries like $\p/\p\beta_{0p}$ or $\p/\p\hat\beta_{0p}$ for the extended KP hierarchy could be reduced to that for the $\hbox{cKP}_{n,1}$ hierarchy.

Similar as in \eqref{Phis}, for arbitrary constants $\kappa$ and $\lambda} \newcommand{\Ld}{\Lambda$ we can define the following equations:
\begin{equation}\label{Phis2}
\frac{\pd \Phi}{\pd s_p'}=-\left(S_p+(\kappa p+\lambda} \newcommand{\Ld}{\Lambda)\hat{P}^{p}\right)_-\Phi, \quad
\frac{\pd\hat{\Phi}}{\pd s_p'}=\left(S_p+(\kappa p+\lambda} \newcommand{\Ld}{\Lambda)\hat{P}^{p}\right)_+\hat{\Phi}, \qquad p\ge-1.
\end{equation}
In the same way as above, by doing the replacement $S_p\mapsto S_p+(\kappa p+\lambda} \newcommand{\Ld}{\Lambda)\hat{P}^{p}$, we achieve that the flows $\p/\p s_p'$ are consistent with the constraint \eqref{PnPh}, and they are reduced to
\begin{equation}\label{LsL}
\frac{\p L}{\p s_p'}=\left[-(S_p+(\kappa p+\lambda} \newcommand{\Ld}{\Lambda)\hat{P}^{p})_-, L\right], \quad p\ge-1.
\end{equation}
Moreover, such reduced flows satisfy:
\begin{align}\label{}
\mathrm{(i)} \qquad & \left[\frac{\p}{\p t_k}, \frac{\p}{\p s_p'}\right]L=0,  \\
\mathrm{(ii)} \qquad
& \left[\frac{\p}{\p s_p'}, \frac{\p}{\p s_q'}\right]L=(q-p)\frac{\p L}{\p s_{p+q}'},
\end{align}
where $k\in\Z_{>0}$ and  $p, q\in\Z_{\ge-1}$.

\begin{rmk}
The reduced flows defined by \eqref{Phis} and by \eqref{Phis2} under the constraint \eqref{PnPh} satisfy
\begin{equation}\label{LsLsL}
\left(\frac{\p}{\p s_p'}-\frac{\p}{\p {s}_p} \right) L=\left\{
\begin{array}{cc}
  0,  & p=-1, 0; \\
 -(\kappa p+\lambda} \newcommand{\Ld}{\Lambda) \left[(L^p)_-, L\right], & p\ge1.
\end{array}\right.
\end{equation}
\end{rmk}

It is known that, for the $\hbox{cKP}_{n,1}$ hierarchy \eqref{Lt}, a series of the Virasoro symmetries was constructed by Aratyn, Nissimov and Pacheva in \cite{ANP} by adding certain  ``ghost'' symmetry flows related to the eigenfunctions characteristic for the operators $L$ and $L^*$ (in fact more general cases were studied there). More exactly, in Proposition~2 of \cite{ANP} the following Virasoro symmetries were given:
\begin{equation}\label{Lstil}
\frac{\p L}{\p \tilde{s}_p}=\left[-\frac{1}{n}(M P^{n p+1})_- +Y_p , L\right], \quad p\ge-1,
\end{equation}
where $Y_{p} =0$ for $p=-1,0,1$, and
\begin{equation}\label{Yp}
Y_{p} =\sum_{j=0}^{p-1} \frac{2 j-p+1}{2}L^{p-j-1}(v)\p^{-1} (L^*)^j(w), \quad p\ge2.
\end{equation}
Here we repeat
\[
v=e^f=\hat{\Phi}(1), \quad w=\rho e^{-f}=\left(\hat{\Phi}^{-1}\right)^*(1),
\]
and note that there are nonlocal-action terms in $Y_p$ whenever $p\ge2$.

\begin{prp}\label{thm-ANP}
The flows defined by \eqref{Lstil} satisfy
\[
\frac{\p L}{\p\tilde{s}_p}=\frac{\p L}{\p s_p'}, \quad  p \ge-1,
\]
where ${\p}/{\p s_p'}$ are given  in \eqref{LsL} with $\kappa=-\frac{1}{2}$ and $\lambda} \newcommand{\Ld}{\Lambda=\frac{1}{2}$.
\end{prp}
\begin{prf}
Let us fix $\kappa=-\frac{1}{2}$ and $\lambda} \newcommand{\Ld}{\Lambda=\frac{1}{2}$ in \eqref{LsL}, then
\begin{align*}
\left(\frac{\p}{\p\tilde{s}_p}-\frac{\p}{\p {s}_p'} \right) L
=&\left[-\frac{1}{n}\left(M P^{n p+1}\right)_- +Y_p+\left(S_p-\frac{p-1}{2}\hat{P}^p\right)_-, L\right] = \left[Y_p+Z_p, L\right],
\end{align*}
where
\begin{equation}\label{Zp}
Z_p:=\left(\hat{M}\hat{P}^{p-1}-\frac{p-1}{2}\hat{P}^p \right)_-=\left(\hat{\Phi}x \p^{-p+1}\hat{\Phi}^{-1}\right)_- - \frac{p-1}{2}\left(\hat{\Phi}\p^{-p}\hat{\Phi}^{-1}\right)_-.
\end{equation}
When $p=-1,0,1$, clearly $Z_p=0=Y_p$, then the flows $\p/\p\tilde{s}_p$ and $\p/\p s_p'$ acting on $L$ coincide. When $p=2$, on one hand from \eqref{Yp} we have
\begin{align*}
Y_2=&\frac{1}{2}\left(-L(v)\p^{-1}w+v\p^{-1}L^*(w)\right) \\
=&-L(v)\p^{-1}w+\frac{1}{2}\left(L(v)\p^{-1}w+v\p^{-1}L^*(w)\right) \\
=&-L(v)\p^{-1}w+\frac{1}{2}\left(L^2\right)_-,
\end{align*}
where the last equality is due to the formula (61) in the appendix of \cite{ANP}, say, in the present case
\[
(L^k)_-=\sum_{j=0}^{k-1}  L^{k-j-1}(v)\p^{-1} (L^*)^j(w), \quad k \ge1.
\]
On the other hand, since
\begin{align*}
\left(\hat{\Phi}x \p^{-1}\hat{\Phi}^{-1}\right)_- =& \hat{\Phi}(x)\cdot \p^{-1} \cdot\left(\hat{\Phi}^{-1}\right)^*(1) =\hat{\Phi}\p^{-1}\hat{\Phi}^{-1}\hat{\Phi}\p(x)\cdot \p^{-1}\cdot \left(\hat{\Phi}^{-1}\right)^*(1)  \\
=&L\hat{\Phi}(1)\cdot\p^{-1}\cdot w=L(v)\p^{-1}w,
\end{align*}
then from \eqref{Zp} we have
\[
Z_2=L(v)\p^{-1}w-\frac{1}{2}\left(L^2\right)_-.
\]
Clearly $Y_2+Z_2=0$, then we obtain $\p L/\p\tilde{s}_2=\p L/\p s_2'$.  When  $p\ge3$, the flows $\p/\p \tilde{s}_p$ and $\p/\p \tilde{s}_p'$ can be determined by those flows with $p=-1,0,1,2$ via the Virasoro commutation relations. Therefore we complete the proof.
\end{prf}

\section{Concluding remarks}

In this paper we have represented the extended KP hierarchy into a bilinear equation satisfied by the (adjoint) Baker--Akhiezer functions. What is more, we have constructed a class of additional symmetries for this hierarchy, and studied their reduction properties. Such results, which are analogue to those for the KP and the 2-BKP hierarchies, are expected to extend our understanding to the theory of integrable systems.

At the end of \cite{WZ}, a tau function of the extended KP hierarchy was introduced by using the densities of Hamiltonian functionals. Similar to the KP hierarchy, one can derive a Sato formula that links the tau function to the (adjoint) Baker--Akhiezer functions $\psi(z)$ and $\psi^\dag(z)$. However, the relationship between the tau function and $\hat{\psi}(z)$ (or $\hat{\psi}^\dag(z)$) still needs to be clarified. It is why a bilinear equation of tau function is still missing. We also hope that the tau function of the extended KP hierarchy could be applied in the two-matrix model such as in the context of constrained KP hierarchies \cite{ANP-DB}. This will be studied elsewhere.

\vskip 0.5truecm \noindent{\bf Acknowledgments.} {The authors thank Professor Baofeng Feng for helpful discussions. This work is partially supported by the National
Natural Science Foundation of China Nos.\,12022119, 11771461, 11831017 and 11521101.

{\small

redpajama_set_name: RedPajamaArXiv


Modern purpose built apartment with panoramic sea views, having two double bedrooms, lounge/fitted kitchen/dining room, bathroom, shower/cloakroom, utility, under floor central heating, uPVC double glazing, balcony and level parking space. Viewing is strongly recommended - Chain free.
Personal entrance door. Audio door entry system. Doors into two bedrooms, shower/cloakroom, utility and living room.
Modern fitted white fronted kitchen units comprising four wall cupboards, five base units and one drawer, with roll top work surface, breakfast bar, inset stainless steel sink and drainer with mixer tap over, built in electric oven with four ring hob and extractor over. Integrated dishwasher and refrigerators and freezer.
UPVC double glazed window with sea views. Radiator. Door into en suite bathroom.
Incorporating panelled bath with glazed shower screen and shower over, low level flush WC and wash hand basin. Centrally heated ladder style towel rail. Extractor fan.
UPVC double glazed sliding patio doors gaining access to enclosed balcony. Radiator.
Incorporating double shower with glazed sliding doors, low level flush WC and wash hand basin. Central heated ladder style towel rail. Extractor fan.
There is a level block paved parking space for one car.


Vegan coleslaw : Road to healthy eating , packed with greens and flavor.
Coleslaw could be made by skipping the mayo that makes it egg free but without the creaminess and tartness of it binding the veggies it becomes just like any other salad. I have been trying to replace mayo with sour cream, buttermilk and what not. But thats when it struck me how yogurt and coconut is used together for creamier texture in my native curries which makes it pure vegetarian .
Mung bean sprouts add a nutty flavor and crispness to the slaw.
Now vegan coleslaw is a complete protein rich and high fiber well balanced diet.
1. Cut the apple into thin long strips , squeeze lil lime juice on it and keep it aside.
2.Mix the dressing ingredients well.
3. Into a large bowl put all the veggies and pour the dressing as required and mix well. Refrigerate for an hour until flavors develop.Then enjoy!
Note : if you like the cabbage slightly wilted then give a firm squeeze to the cabbage with ur hand before adding the rest of the veggies.
Serve as a salad for vegans and for all the carnivores :)) out there serve along with spicy fish, chicken or beef etc..
This dish is sent to MLLA#73 hosted by Food for 7 stages of life.
Love the vegan take on this dush and love sprouts in it.
I like the idea of Vegan dressing.. 🙂 I don't like mayo and so this is something I would like to try on. Lovely pics!
Triple yes to this awesome salad. Ahhh!! the pictures are equally fabulous.
I missed reading vegan…omg, I scrolled back to see the recipe..I literally love the dressing recipe..
Hi Meena, that salad dressing is simple, but amazing. loved the healthier spin.
Vegan coleslaw looks delicious I will make it next time when I buy cabbage. Usually I make thoran with it.
Thanx for the recipe dear..will dfntly gve it a try..


Some of you are wondering how to use Windows successfully while knowing about the advantages of a Linux workstation. To clarify a bit I created a plain, curated list. You are welcome to comment and recommend useful (free or open source) tools.
First I'd like to have an apt-like application installer. This is where chocolatey Scoop steps in.


text:
The Negation of Humanity
And in Mexico?
Classifying, Ordering, Dominating…
Miscegenation and Identity
The Colonial Order
The Order of a New Nation
The Contemporary Order
The Dignifying Gaze
Discrimination due to skin color, physical features and one's way of speaking and dressing is a fact of daily life in Mexican society, as it is in many other parts of the world. The terms that accompany this discrimination – indio patarrajada, naco, chaca and many others – show how these physical and cultural characteristics are assigned a set of values; that is, it is thought that white skin is better than dark skin, narrow noses better than broad noses and Spanish better than Nahuatl or Maya.
Let's Talk About Racism, 2016, Video Installation. Idea and text by César Carrillo Trueba. Produced by La Maga Films
See full room...
Of the many forms of discrimination that are typical of the contemporary world, racism is distinguished by its basis in the differences between different human groups, whose features – such as skin color or hair shape – have been defined as constituting a race and are then converted into signs of inferiority or superiority. A clear example of this can be seen in the history of the United States, where slavery was based on the idea of the black inferiority, an idea that persisted after its abolition in 1864 with the passage of laws that prevented them from using public spaces, enrolling in college and, up until a few decades ago, requiring them to cede their seat on the bus to whites. The repeal of these laws did not end racism, whose institutional expressions became more subtle, but it nevertheless remains intense in its social manifestations.
Tags: La humanidad negada
Tags: Y en México
The word "race" began to be widely used in classifications of human beings starting in the second half of the 19th Century, contemporary to the rise of European expansionism and the founding of powerful empires. The attributes conferred upon each race speak to the system of relationships established at that time, in which Europeans defined themselves as the most civilized, the most highly evolved, those residing in the most favorable part of the world: in sum, the most perfect race, "the most beautiful and best-made men," as the celebrated French naturalist Buffon said.
Hipólito Salazar. Allegories of the Four Continents, 1866. Lithograph. INBA/MUNAL Archives.
Hipólito Salazar Alegorías de los 4 continentes, 1866 Litografía Acervo INBA/MUNAL
Tags: Clasificar ordenar
How do you determine which features define a race? By measuring, quantifying, making statistical generalizations, separating and grouping populations to give greater weight to one characteristic or another, such as skin color or hair shape. In this desire to provide a scientific basis, racial studies began to utilize special methods, instruments and instructions, giving rise to enormous inventories of nose shapes, eye colors, head sizes and brain sizes, but also of behaviors, nutritional patterns, moral and political systems and many other cultural and social traits.
Luis González Palma. The Critical Gaze, 1999, Hand-Colored Silver on Gelatin, La Penca Producciones Transdiciplinarias A.C. Collection.
Tags: El Rostro
The division of humanity by skin color is the most common and widespread classification, but also the most arbitrary, as it's difficult to argue that Africans are truly black, Asians truly yellow, Europeans truly white – and even more so, to establish where one skin tone begins and another ends, as it's a physical trait that exists on a continuum, and can even vary in a single person due to light conditions and exposure to the sun, among other factors. If to this we add the value attributed to this trait – the supremacy of white skin above all others – the description and judgements of the skin of others and the perception of one's own appear in all their complexity.
Unknown Artist, Castes, 18th Century, Oil on Canvas, Archives of the National History Museum, Chapultepec Castle, INAH.
The concept of race took on such importance at the end of the 19th Century and the beginning of the 20th that the interpretation of history and society and their evolution was based around it. The way in which the foundational myth of the newborn nation was written following the Mexican Revolution of 1910 forms part of this line of thinking. As in many other parts of the world, two visions contrasted and complemented each other: one praised racial purity and considered miscegenation to have disastrous effects (which led Germany to Nazism), and the other saw miscegenation as the ideal way to elevate a race by mixing it with another that was considered to be more advanced – thus "improving the race," as was the case in Mexico.
Humberto Limón, Crucible of Races, 1975, Oil on Canvas, Soumaya Museum Collection.
I. Conquest and Genocide, 2016, Charcoal on Newsprint on Wood, La Penca Producciones Transdiciplinarias A.C. Collection.
Racism – like any form of discrimination – implies a mental order and a spatial order. In the eyes of the Spaniards who explored America, indigenous people constituted an enigma: they asked themselves if they were human. Many doubted it, believing them to be closer to beasts than men. The Dominican friar Tomás Ortiz wrote that "they are incapable of learning [...] God has never created a race full of more vices [...] Indians are stupider than donkeys and they reject all progress."
César Rangel, Civilizing Triptych II: The Genocide of the Yaqui People, 2016, Charcoal on Newsprint on Wood, La Penca Producciones Transdiciplinarias A.C. Collection.
In 1821, the Plan of Iguala stipulated that "all inhabitants of New Spain, whether they be Europeans, Africans or Indians, are citizens of this monarchy with the same opportunities for employment, depending on their merits and virtues," thus laying the groundwork for equality in the newborn Mexican nation. Nevertheless, difficulties soon arose in making this a reality, as the idea of imposing a European way of life, of being, was to clash with a majority-indigenous population – what was called the "Indian problem," an obstacle to so-called progress.
César Rangel, Civilizing Triptych III: Contemporary Racism, 2016, Charcoal on Newsprint on Wood, La Penca Producciones Transdiciplinarias A.C. Collection.The ideal of miscegenation promoted by the revolutionary government saw the city as the perfect laboratory for this experiment: a raceless place where there would be neither whites nor indigenous people, allowing everyone to be a mestizo citizen enjoying equal rights. As if responding to this call to assimilate, to leave behind their culture, their language and their way of life, many did go, spurred on by the poor conditions in the countryside and seeking out jobs and a new way of life.
Indigenous people tend to be represented in very specific ways, varying between the picturesque, paternalism, folklore and clearly unfavorable representations. Non-indigenous people are portrayed using different criteria. What happens when people from both groups are portrayed the same way? Barriers disappear and it becomes difficult to know who's what; there's a distancing effect, and we don't recognize those whom we see in our everyday surroundings. Our perception is freed from the references that fetter it and equality emerges in all its diversity. By breaking with established clichés, the gaze of the artist dignifies everyone. There's no longer a place for discrimination. It's the beginning of the abolition of racism.
Margot Sputo, Indigenous/Non-Indigenous, 2015-2016, Photography, La Penca Producciones Transdiciplinarias A.C. Collection.
Project and Curatorship: César Carrillo Trueba (La Penca Producciones Transdiciplinarias A.C).
Coordination: Natalia Gabayet and Delphine Kachadourian
Administration: Daniela Cruz Benhumea, Delphine Kachadourian and Elena Coll
Special Assistance: Josué Ramírez
Research and Texts: César Carrillo Trueba
Archival Research: Itzel Ávila and Elizabeth Balladares
Reprography: Agustín Estrada
Photo Printing: Agustín Estrada, Jorge López, Imágenes Vanguardista

redpajama_set_name: RedPajamaCommonCrawl


 How Enterprise Mobility Can Grow Productivity in Manufacturing Industry?
How Enterprise Mobility Solutions Can Increase Productivity in Manufacturing Industry?
If service sector can implement enterprise mobility solutions to strengthen their business, can manufacturing sector be far behind?
With sincere apologies to the great romanticist Shelley, the point being made is quite clear.
Almost over 80% manufacturing institutions think that enterprise mobility solutions for manufacturing do give a considerable boost to their business. In the same way, over 70% of employees believe that such solutions incredibly increase their productivity while helping them maintain work-life balance.
Businesses can get mobility solutions for manufacturing in a variety of manufacturing ranging from digital manufacturing to distribution ecosystem.
BlackBerry conducted a study to note the actual effect of BlackBerry mobility solutions. As per that study, an employee who used a smart phone has been able to increase her productivity by one hour that was a benefit of $12,500 per user on the average annual salary of $100,000.
The same study also noted that workflow efficiency increased by 38% which directly resulted into company's gain of $33,630 per employee annually. In terms of immediate communication, mobility solutions for manufacturing deployed by BlackBerry enabled employees to send and receive 24 e-mails per day among which 9 were time-sensitive. Considering the traditional value of $5 of each time-sensitive e-mail, the company benefited $4,400 per user annually. Overall, the mobility solutions enabled the company to gain $50,000 per employee annually.
What area of manufacturing can be the first one to go mobile?
What should be the ideal road-map?
Can mobility solutions for manufacturing be controlled by single-point of control?
How to integrate mobility solutions with the rest of the business?
While all these challenges are genuine, they can be easily overcome if you collaborate with a technology firm such as Space-O. Space-O, apart from offering enterprise mobility solutions for manufacturing industry also offers consultations.
If you are into any type of manufacturing business, Contact Space-O right now and know how enterprise mobility solutions for manufacturing industry can change the way you do business.
This page was last edited on February 22nd, 2019, at 9:18.
Why Web Scraping is Important for Large Websites (Airbnb, Flipkart…) in this Technologically Driven World?


knowledge, experience and skill to your project, assisting you to meet your required aims and outputs.
The BPM Marine Turtle Team complements the existing extensive marine megafauna expertise within BPM. Drawn together by BPM and comprised of five key members (each an experienced and independent specialist in marine turtle research), the Marine Turtle Team works with us on a project-by-project basis. This collaborative approach provides BPM with the flexibility to adapt to the specific needs of each project and provides you with the confidence that your project is in the hands of the right expert. In addition, we have fostered relationships with a wide international network of marine turtle specialists, so when required, we can easily complement the skills of our key members.
BPM will work closely with you at all times in order to meet your environmental, conservation, management and social goals. All of our work and systems are subject to quality standards based upon ISO 9001, AS/NZS 4801 and ISO 14001.
The five key members of the BPM Marine Turtle Team have been carefully selected for their extensive and complementary knowledge, experience and skills. Together, they have worked with all species of marine turtles found in Australian, New Zealand and South Pacific waters, and have exceptional survey design, fieldwork, analysis, reporting and communication skills.
The members of the Marine Turtle Team have made significant contributions to the scientific knowledge about marine turtles. Together they have 85 years of experience working with marine turtles, and published over 30 peer-reviewed scientific papers, two book chapters and 42 government reports on marine turtle ecology, biology, genetics, conservation and management. They have also dedicated time to capacity building projects within local communities.
Dr Nancy FitzSimmons is Australia's leading marine turtle geneticist. Since 1991 she has conducted research on all species of marine turtles in the Indo-Pacific in order to define breeding populations, delineate Management Units and determine stock composition of feeding ground populations. She has worked closely with state and federal government to convey this information and guide policy development. In 2013 Nancy co-authored the Australian Commonwealth Government's Draft Marine Turtle Recovery Plan.
Barrier Reef green turtle population for the Great Barrier Reef Marine Park Authority. This user-friendly model is used by the Authority and Indigenous ranger groups to assess the severity of impacts upon this population of green turtles. Nancy has considerable experience in the training of volunteers (fishermen and community members) for local nesting monitoring and stranding programs, and in the development and delivery of training workshops for turtle nest monitoring activities in order to build capacity within indigenous communities. She has extensive involvement in eco-tourism/turtle programs. As an active member of the IUCN marine turtle specialist group since 1998, Nancy has provided information on marine turtle stock structure for the IUCN, the United Nations Environment Program and the US Government. Nancy is an Adjunct Associate Professor at Griffith University, Australia.
Since 1996 Dr Michelle Boyle has specialised as a Marine Turtle Ecologist and Biologist in Australia, New Zealand, Fiji, New Caledonia and Torres Strait. She has worked within and alongside research institutes, government agencies, NGOs, environmental consulting companies and local communities.
Highly experienced in all aspects of marine turtle nesting research and foraging ground work, Michelle's research has also focussed on the migrations of the pelagic juvenile life history stage of green and loggerhead turtles from east Australia. Her work is multidisciplinary and incorporates spatial and temporal distribution, oceanic currents, diet analysis, and genetic sampling and analysis. Through her work Michelle has collaborated with international researchers, fishermen, indigenous communities, government agencies and NGOs.
She has been involved in regional and community-based consultation, community engagement, training workshops for community capacity building and provided expert scientific advice on anthropogenic issues relating to marine turtles. Michelle has a particular interest in empowering communities to manage their marine turtle resources. She spent 12 months in Fiji where she engaged with local fishermen and families in order to collect information on the harvest of marine turtles. More recently she developed a database for the Torres Strait Regional Authority for use by indigenous rangers when collecting biological data on marine turtles and dugongs harvest. Michelle has extensive involvement in eco-tourism/turtle programs.
In 2013 Michelle co-authored the Australian Commonwealth Government's Draft Marine Turtle Recovery Plan. She has also been involved in undertaking environmental audits on the effects of LNG developments on the nesting turtles. Michelle is an independent Marine Environmental Consultant.
Dr Jason van de Merwe's primary interest is the management of interactions between marine turtles and human activities. His PhD research focussed on chemical contamination of marine turtles, and since 1999 he has worked with government agencies, non-profit organisations, research institutes and environmental consulting companies in Australia, Malaysia the South Pacific and Hong Kong providing technical advice, scientific research, impact assessments and community engagement in marine turtle conservation.
Throughout his work Jason has gained extensive experience in marine turtle nesting studies, foraging ground studies (including rodeo capture), migrations and localised movements (involving satellite telemetry), and collection of blood and tissue samples for the analysis of chemical contaminants. In 2013 he co-authored the Australian Commonwealth Government's Draft Marine Turtle Recovery Plan. Jason is a Research Fellow at Griffith University, Australia.
A Marine Biologist since 2000, Dan Godoy is currently completing his PhD on the biology and conservation management of green turtles in New Zealand. As part of his research, he set up the first marine turtle sighting network and satellite tracking study for marine turtles in New Zealand. In addition, he incorporates turtle necropsy, diet analysis, and genetic analysis to investigate the biology, origin and anthropogenic impacts of green turtles in New Zealand. Dan is experienced in marine turtle field research including turtle nesting surveys and the capture of free ranging turtles. In this area he has worked with researchers, private companies, NGOs, government agencies, and local communities in New Zealand, Australia and Papua New Guinea.
Dan has delivered marine turtle conservation and education seminars throughout New Zealand and in Papua New Guinea, and has provided scientific advice to the New Zealand Government regarding handling, recovery, and emergency firstresponse of marine turtles. Recognising his expertise, in 2013 Dan was elected as a member of the IUCN Species Survival Commission Marine Turtle Specialist Group – Oceania Region. Dan is a PhD candidate at Massey University, New Zealand, and an independent Marine Environmental Consultant.
Paul McCann is a highly skilled Marine Biologist with demonstrated experience in the management, supervision and refinement of scientific marine turtle monitoring programs, experimental design and research field work. Throughout his 12-year career he has amassed a valuable combination of technical and research skills. Paul has worked on marine turtle programmes for industry and non-government organisations in Australia, India and the Seychelles. Projects include: artificial light impact studies on flatback turtle hatchlings; satellite tracking and monitoring studies of flatback turtles incorporating vessel- and shore-based surveys of marine turtle nesting and foraging habitats; environmental assessment and marine baseline studies; nesting, tagging and focal behaviour studies on hawksbill turtles; and data collection and supervision of a hatchery monitoring program for olive ridley turtles. Paul complements his marine turtle expertise with a wealth of experience in HSE management, direction and management of large, multi-disciplinary environmental monitoring programs and EIA. Paul is an independent Marine Environmental Consultant.
implementation of policy, and environmental training.
advice, operational, documentation and strategy support.
equipment, HSE and project management.
advice, documentation, environmental training, collaboration and partnership.
discussion documents, reviews, management plans, strategic and advice papers.
We can also produce educational packages and material.


I've been visiting the Algarve for the past 3 years and part of the appeal for these holidays is doing a whole lot of nothing. We'll spend the morning lazing around the pool and the afternoon going for a little drive and stumbling upon gorgeous beaches. We don't make much of a plan and don't have a big list of 'must-see attractions' or places we simply have to visit.
But over the years we have found a few favourite spots on the Algarve and these are the places I'm sure we'll be returning to next time we visit. Here some of my favourite restaurants, bars, beaches and sights on the Algarve.
This super stylish beach bar has such a cool atmosphere. It's where the beautiful people hang out but it's also really welcoming, has a great menu, really friendly staff, a beautiful beach and it's also child friendly. There's a little play park at the back and they give the kids colouring books to keep them quiet while you enjoy a cocktail.
Go for the deliciously fresh fish but stay for the homey and welcoming atmosphere. This is the kind of place where people arrive at 7pm and stay until midnight. They don't expect you to get up and leave once you've eaten, they actually expect you to order another jug of wine and start chatting to the other people around your table as you enjoy the sound of the waves crashing on the shore.
Beach Bar Burgau overlooks a gorgeous beach. Get yourself a seat on the decking or on one of the sun loungers below and order yourself an ice cold beer. We found this to be a nice place to base ourselves when visiting the beach with kids. Babies and beaches don't go so well together so it was nice to have a shady, comfortable and non-sandy spot!
You can't visit the Algarve without visiting this stunner of a beach. It's an enormous surfing beach backed by sand dunes. A pretty boardwalk down to the beach and to a gorgeous viewing area make it particularly special.
Read more about Praia da Bordeira in my last blog post.
OK, it might not be traditional Portuguese food but it's fresh, healthy, delicious and unusual. I'd describe Micro Bar to be like a vegetarian restaurant that serves meat. It's colourful and healthy and in the loveliest setting in the little town of Carrapateira.
You need to take a boat out to Benagil Caves (accessed from Praia de Benagil – it's well signposted) and you'll be met by this amazing scene.
We were unlucky on the day we tried to visit and the sea was too rough for the boats to leave. I'd recommend trying to visit as early in your holiday as possible so you can give it another go if boats are cancelled. Even if the sea is too choppy, Praia de Banagil is a lovely little sheltered beach so you could stay and enjoy the beach.
Decribed as one of the most beautiful and emblematic beaches on the Algarve, Praia da Marinha is definitely worth a visit! According to TripAdvisor it's the best beach in Portugal and those epic arches make it oh so Instagrammable.
Carvoeiro is the town we stayed in during our first trip to the Algarve and it was probably our favourite spot. The lovely little town is bustling enough to keep you busy without feeling too big and touristy. The beach is really nice and there are lots of restaurants to choose from.
This is one of those places where you can't quite believe that nature did this. Natural carved these enormous rocks into such gorgeous formations. The emerald green waters and soft, golden sand are just the icing on the cake.
This lovely beach bar in Albufeira ticks every box. Stunning location, great food, stylish décor, lovely staff and a beautiful beach! It's also good if you're dining with kids as they can play on the beach and you can see them from your table.
I've only been to the Algarve once but I fell in love with it and know I want to go back for a relaxing break at some point. Some of my favourite places (we were in the Eastern Algarve) were Tavira, the roman ruins at Estoi and Restaurante Baixamar in the small town of Santa Luzia – as well as all the incredible beaches!
Hello, I am planning for Algarve Holidays but surely I would not like to waste time on restaurant and bars. Thanks for great idea about restaurants which might be time saving on holidays but please do add some interesting places to visit.
The beaches pictures are very nice and the Benagil Caves was really interesting.


text:
Two Thirds Of Liverpool Supporters Want The Club To Replace Coutinho This Summer
Author: Content Editor
There has been a lot of speculation regarding Liverpool's transfer targets this summer but it appears that the club are in for a quiet transfer window.
A new left-back and a new forward have been two areas where the club are believed to be considering strengthening, but one position that appears to have been neglected is in midfield.
As we mentioned in a post last week, Liverpool's midfielders do not score many goals, with James Milner scoring the most with just five in the league last campaign.
The feeling among many Reds supporters is the club have never really replaced Philippe Coutinho in the middle of the pitch. When we asked fans whether they want the club to sign an attacking midfielder this summer, two-thirds agreed:
There's no doubt that Liverpool were considering strengthening that area last year when Nabil Fekir was close to signing from Lyon, but things have gone quiet since then. Will they consider a move for another playmaker this summer? We shall see.


text:
At Gallop Racing we will only employ the best trainer. Trainers with a track record of getting the very best out of a horse, looking after the best interests of the horse and with an excellent win to run ratio.
This policy means that our initial monthly costs may be higher as the best trainers command higher training fees. However, it is a policy that pays dividends.. poor (cheaper) trainers don't win very often.
David O'Meara
Ex National Hunt jockey who has set the flat racing training ranks alight with a meteoric rise. From a standing start, David has trained over 500 winners in a 5 year period.
David's ability to improve horses soon became apparent with horses like Pepper Lane. He took a horse rated in the 50's and guided her to consecutive big race successes in the Great St Wilfred and a rating well over 100.
… And so it continues, Penitent, discarded as a 'has been' by previous connections was rejuvenated and improved by David to enjoy Group 2 success.
David has now proven his ability to train to the highest level with 16 group race successes in the last 4 years, including Group 1's for Amazing Maria, Move In Time and G Force.
David has recently moved to a new purpose made yard near York. Set in over 100 acres, with a brand new 6 furlong uphill Safetrack gallop manufactured and installed by Andrews Bowen, Willow Farm is the perfect place to train racehorses.
Clink on this link to view David's Facilities https://www.davidomeararacing.com/about/willow-farm
VIEW DAVID'S STATISTICS
Mark Johnston
Mark Johnston is one of Britain's leading racehorse trainers, having become the first Flat trainer to send out more than 200 winners in a season (2009), a feat which he has, to date, repeated in 2010, 2012, 2013, 2014, 2015, 2017, 2018 and 2019. On August 23, 2018, Mark became the winning most trainer in British racing history when he saddled his 4,194th winner.
Uniquely, he has trained more than 100 British winners for 26 consecutive seasons, achieving success in British and Irish Classics and in numerous Group One races.
VIEW MARK'S STATISTICS


text:
Q: Apply Function on all rows to get a matrix output in R I need to apply a function on all the rows of the dataframe. This parameters for these function will be two vectors,
For example,
f <- function(vec1,vec2)
     { return(sum(vec1) - sum(vec2))
     }

Now I need to apply this on all the rows of my dataframe, so that I can populate all values in a matrix and use that matrix for my analysis.
df <- data.frame(A=c(1,2,3,4),B=c(3,4,3,4),C=c(5,6,4,5))

Applying this function should give the following matrix

A: Try this:
m <- diag(nrow(df))
s <- rowSums(df)
r <- combn(nrow(df), 2, function(x) c(s[x[1]]-s[x[2]],s[x[2]]-s[x[1]]))
m[lower.tri(m, diag = F)] <- r[1,]
m <- t(m)
m[lower.tri(m, diag = F)] <- r[2,]
diag(m) <- 0

     # [,1] [,2] [,3] [,4]
# [1,]    0   -3   -1   -4
# [2,]    3    0    2   -1
# [3,]    1   -2    0   -3
# [4,]    4    1    3    0

In case you insist to use you own defined function f, do this:
m <- diag(nrow(df))
r <- combn(nrow(df), 2, function(x) c(f(df[x[1],],df[x[2],]), f(df[x[2],],df[x[1],])))
m[lower.tri(m, diag = F)] <- r[1,]
m <- t(m)
m[lower.tri(m, diag = F)] <- r[2,]
diag(m) <- 0

redpajama_set_name: RedPajamaStackExchange


VOS SIGNAGE design, print and install Graphics and commercial ads vinyl decals on vehicles such as: cars, trucks, SUVs, trailer homes, working trailers, motor homes, and much more.
Printed vinyl and cut-color vinyl for vehicles has great resistance to outside weather with specific design application suited to moving vehicles.
Printed see-through vinyl, also used for vehicle windows, with protected laminate film or UV-treated ink up to 5 years.
Why vehicle graphics and decals?
• Direct guide for business and company identities.
• Great commercial marketing ads message for business.
• It's easy to install & easy to be removed.
• No license requirements in North American cities.
• It's affordable and gives fast return investment.


text:
FIELD AND GREENHOUSE VEGETABLE PRODUCTION ON PEAT AND BROWN COAL DUST
E. Chroboczek, J. Dobrzanska, O. Nowosielski, J. Rumpel
Poland has about 1. 2 milion hectares of peat, 95% of which is highly decomposed peat or muck, suitable for field production of various plants and also vegetables. The Research Institute of Vegetable Crops in Skierniewice has an Experiment Station at Rekowo at the Baltic coast near Gdansk, where field experiments on peat soils are being carried on. The peat experimental field of this Station is a part of a 4000 ha peat area "Mos1cie Blota". Mr. J. Rumpel, one of the authors of this paper, conducted there field experiments for three years (1968–1970) with onions, sown directly in the field, with transplanted onions, started in hotbeds, with cabbage and with carrots, on soils treated with different kinds and amounts of macro- and micro-elements.
The peat soil of these experiments had the pH value (in water) of 6. 1 and the degree of its decomposition was 74%. The peat layer is 1. 85 m deep. Nitrogen in the form of ammonium nitrate was applied on all plots in equal amounts of 50 kg N/ha. Phosphorus and potassium were applied in two different doses, one being 3 times the other. The single dose of phosphorus was 80 kg of P2O5 in the form of the 18% superphosphate and that of potassium was 160 kg of K2O ha in the form of potassium sulfate. Copper in the form of copper sulfate in the amount of 100 kg/ha was used evenly on all plots. Molybdenum, also the subject of the experiment was applied on some plots. The addition of manganese to the peat was not necessary, because chemical analyses and the previous pot experiments have shown that the amount of this microelement in the soil was sufficient for the plants. Every fertilizer treatment was repeated four times. Liro - CIPC was used as a herbicide in onions (twice) and Gesagard in carrots.
Experimental results for onions, sown in the field, are presented in table 1. For clarity we are giving here results only from the first year of the experiment and the average yields for the years 1968, 1969 and 1970. The size of plots harvested was 12 m2, but the yields in table 1 are given in quintals per hectare.
The figures in table 1 show, that the highest yield in 1968 (total and marketable) was obtained with Lot 6, i. e. from plots fertilized with a triple amount of phosphorus (N 3 P K Cu Mo). This lot brought the highest yield in every year of the experiment, as is evident from the average yields for the three year period. The figures of yields have been analysed statistically (Duncan test with P = 0. 95). The difference in yields between Lot 6 and Lot 2, with a single dose of phosphorus, proved to be highly significant. From the comparison of the yields of the Lots 5 and 6 it is evident that molybdenum was also essential for onions, grown on
Chroboczek, E., Dobrzanska, J., Nowosielski, O. and Rumpel, J. (1972). FIELD AND GREENHOUSE VEGETABLE PRODUCTION ON PEAT AND BROWN COAL DUST. Acta Hortic. 26, 51-60


text:
Live and Become
Preview the movie in Low-Def:
Starring: Yaël Abecassis; Moshe Agazai;Moshe Abebe;Yitzhak Edgar;Roschdy Zem
Director: Radu Mihaileanu
In 1984, an operation by the Israeli secret service airlifted thousands of Falashas, Ethiopian Jews displaced by civil war and famine, to seek solace and refuge in Israel. An Ethiopian Christian widow urges the mother of a deceased boy to take her 9-year-old son and instructs him to 'become' Jewish in order to survive. Once in Israel, he encounters isolation and loneliness as he longs for his mother and stumbles in a society that does not fully accept him. The film recounts his journey as a young boy, a teenager and a fully grown adult as he learns to live with the drastic changes of his adopted home. Comming Soon!
Anteregnaw
The Other side of Love
ADERA Free Preview


text:
HIRO-X (Eredeti neve: Aoki Hiromicu, Sizuoka prefektúra, 1980. december 26. –) japán énekes. A Prince of Tennis anime első nyitódalával debütált. Annak ellenére, hogy a dal sikeres és közkedvelt volt, szólókarrierje során csupán 5 kislemezt jelentetett meg. Ebből 4 a Prince of Tennis animéhez köthető, míg a THE MEANING OF TRUTH az F-Zero Falcon Denszecu anime nyitódala. Jelenleg két indie együttes, a SHANDY SCOPE és az akashic note tagja.

Kislemezek
 2001. december 5. – future
 2002. május 1. – Driving Myself
 2003. december 3. – THE MEANING OF TRUTH
 2005. október 5. – Calling
 2011. december 7. – future -reboot version-

Linkek
 Hivatalos weboldal
 SHANDY SCOPE hivatalos weboldala
 Oricon profil

Japán énekesek
1980-ban született személyek
Élő személyek

redpajama_set_name: RedPajamaWikipedia


Our Swimwear Trends are Ahead of the Trends. For salt I use tamari or soy sauce, salt, or miso. Don't see what you're looking for? Forever21 undies are very nice and comfy, especially the lace undies I just wish it came in more color options and prints. Over the years we have worked closely with many brands helping them correct fit issues with the feedback received from our customers.


text:
import type {CourseType, TermType} from './types'
import flatten from 'lodash/flatten'
import * as storage from '../storage'

export async function loadCachedCourses(): Promise<Array<CourseType>> {
	const terms: Array<TermType> = await storage.getTermInfo()
	const promises = terms.map(term => storage.getTermCourseData(term.term))
	const coursesByTerm = await Promise.all(promises)
	return flatten(coursesByTerm)
}

redpajama_set_name: RedPajamaGithub


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Activities on Identiplant are fully logged. Records are kept of when you access the site, courses and resources along with a log of all online communication such as forum posts, assignment submissions, page views etc. These records are available to course administrators and system administrator.
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The right to be informed. You have the right to be provided with clear, transparent and easily understandable information about how we use your information and your rights. This is why we are providing you with the information in this policy.
The right of access. You have the right to obtain access to your information, which you do when logged on to Identiplant. This is so you are aware and can check that we are using your information in accordance with data protection law.
The right to erasure. You can request the deletion or removal of your information where there is no compelling reason for us to keep using it.
The right to data portability. You have rights to obtain and reuse your personal data for your own purposes across different services. For example, if you decide to switch to a new provider, this enables you to move, copy or transfer your information easily between our IT systems and theirs, safely and securely without affecting its usability.
Identiplant has implemented technology and security features, as well as strict internal guidelines, to safeguard the privacy of your personal information from unauthorised access or improper use. We will continue to enhance our security procedures as new technology becomes available.
You can play an important role in keeping your personal information secure by maintaining the confidentiality of your password used on the site. We recommend that you change your password from time to time for additional security. Please advise us immediately if there is any unauthorised use of your account by any other internet user or any other breach of security.
Identiplant is committed to protecting your personal information. We will not disclose your personally identifiable information to anyone outside Identiplant.


[ruby-cvs:49745] akr:r42603 (trunk): * bignum.c (rb_big2str1): Make	an expression more explicit.
Previous message: [ruby-cvs:49744] akr:r42602 (trunk): * bignum.c (rb_big2str1): Use	power_level instead of bitsize(xn).
* bignum.c (rb_big2str1): Make an expression more explicit.


The charter by Van Oord Offshore Wind Projects started on the 5th of January 2015.
The Geo Focus has been the main Survey Vessel during the entire project.
Multibeam, Side Scan Sonar, ROV touch down monitoring during cable laying, UXO and Seismic.
Gemini is located 85 kilometres offshore.
With an installed capacity of 600 megawatt (150 x 4 megawatt wind turbines), Gemini is one of the largest OWF's on the world.
Both in terms of size and production.
Geo Plus is very proud of having been part of this amazing project and also on the trust we have received from our client Van Oord.
The project proved to be a successful cooperation between all parties involved.