text:
\begin{document}

\title{Virasoro symmetries of Multi-Component Gelfand-Dickey systems
}

\author{
\  \  Ling An, Chuanzhong Li\footnote{Corresponding author:lichuanzhong@nbu.edu.cn.}\\[4pt]
\small School of Mathematics and Statistics,  Ningbo University, Ningbo, 315211, China\\[4pt]
}

\date{}

\maketitle

\abstract{In this paper, we mainly study the additional symmetry and $\tau$ functions of a multi-component Gelfand-Dickey hierarchy which includes many classical integrable systems, such as the multi-component KdV hierarchy and the multi-component Boussinesq hierarchy. With other kinds of reductions, we can derive a B type multi-component Gelfand-Dickey hierarchy and a C type multi-component Gelfand-Dickey hierarchy. In our research, the additional flows of the additional symmetries can not all survive. By calculating, we find that the generator of the additional symmetry of the C type multi-component Gelfand-Dickey hierarchy is different from that of the B type multi-component Gelfand-Dickey hierarchy, while the forms of their surviving additional flows are the same.
}\\

{\bf Mathematics Subject Classifications (2010)}:  37K05, 37K10, 35Q53.\\
{\bf Keywords:} multi-component Gelfand-Dickey hierarchy, additional symmetry, string equation, $\tau$ function, Virasoro canstraint.  \\
\allowdisplaybreaks
\setlength{\baselineskip}{12pt}
\tableofcontents

\section{Introduction}

\ \ \ \ Gelfand-Dickey hierarchy was introduced by Gelfand and Dickey\cite{1}, which attracted many people's attention, and then became one of the hottest topics in classical integrable systems. The Hamiltonian theory of the Gelfand-Dickey hierarchy was developing gradually in terms of the Lax pairs, and has become a powerful supports for the study of integrable systems. These theories were introduced in detail in \cite{20}. In fact, many domestic and foreign scholars have done  extensive research on the soliton solutions of the Gelfand-Dickey hierarchy, additional symmetry,  $\tau$ function, B$\ddot{a}$cklund transformation and other related properties\cite{2}. In addition, a lot of research  has been done on supersymmetric Gelfand-Dickey hierarchy,  q-deformed Gelfand-Dickey  hierarchy, etc \cite{3,4,5,6,7,8,11}. Generally speaking, little research has been done on the multi-component Gelfand-Dickey (mcGD) hierarchy. In this paper, the additional symmetries and $\tau$ functions of a multi-component Gelfand-Dickey hierarchy, a B type multi-component Gelfand-Dickey (mcBGD) hierarchy and a C type multi-component Gelfand-Dickey (mcCGD) hierarchy will be studied based on the work done by domestic and foreign  scholars on the KP hierarchy and GD hierarchy.

When the theory of additional symmetries first appeared, it ranked only at the edge of integrable systems. With the continuous study of string equations\cite{12,13,14}, Virasoro constraints\cite{15,16} and other theories, it was found that the additional symmetries played an extremely important role in these theories, and the additional symmetries gradually attracted people's  attention\cite{17,18,19}. A direct application of the additional symmetry of the mcGD hierarchy is to derive the string equation which appears in the study of the string theory. The general expression of the string equation is $[P, Q]=1$, where $P$ and $Q$ are differential operators. For string equation, one of its important characteristics is that it has a close relationship with hierarchies of some integrable equations. The relationship between them is that the string equation is invariant under the flows generated by the equation in the hierarchy\cite{20}.
A portion of the additional symmetries can be reduced to Virasoro symmetries. Virasoro symmetries have a wide range of applications,
of which their roles in $\tau$ function are particularly important\cite{9,10}, because $\tau$ function appears as a partition function or as a generating function in modern problems of mathematics and physics. According to the action of Virassoro symmetries on the $\tau$ function, we can also get the explicit solutions of the Virasoro constainted nonlinear equations in the form of matrix integrals. Such as \cite{22,24}, it was the Virasoro costrainted solution in the Toda chain case and in the KdV case. The additional symmetries can also be used to find the eigenfunctions of the linearized problem and to solve the stability problems\cite{21}.

In this paper, the definitions and properties of a mcGD hierarchy, a mcBGD hierarchy and a mcCGD hierarchy are described respectively. The order of the article is as follows: Firstly, we define their Lax equations by the operator $L$, operator $R$ and appropriate constraints, and introduce the wave operator $\phi$ and discuss some properties of wave functions which naturally lead to their Sato equations. Then, by introducing an Orlov-Shulman's operator $M$, the definition of additional symmetries is given, and many practical properties are derived from the additional symmetries. In the calculation, we find that only a small part of the additional flows can survival, so we give the forms of the surviving additional flows. At the same time, a special additional flow is analyzed and calculated, and an important application of additional symmetries in the string theory is obtained. Finally, through the existence theorem of $\tau$ functions, the additional symmetries of $\tau$ functions are discussed, and the equivalent form of the string equation, Virasoro constraint, is obtained. It should be noted that the mcCGD hierarchy can be divided into odd and even forms.

\section{Multi-Component Gelfand-Dickey hierarchies}

\ \ \ \ The Gelfand-Dickey hierarchy is one of the most important topic in the area of classical integrable systems. The definition of the multi-component Gelfand-Dickey hierarchy is based on a $N$-order differential operator $L$ and a operator $R_{\alpha}$ like these\\
\begin{equation}
\ \ \ \ \ \ \ \ \ \ L=A\partial^{N}+u_{2}\partial^{N-2}+u_{3}\partial^{N-3}+\cdots +u_{0},
\end{equation}
among them, $A=diag(a_{1},a_{2},\cdots ,a_{n})$, $a_{i}$ are nonzero constants,
The diagonal elements of $u_{0}$ are all equal to zero, except in the case of $n=1$. And the $u_{i}$ is an arbitrary $n\times n$ matrix;\\
\begin{equation}
R_{\alpha}=\sum\limits_{i=1}^{\infty}R_{i\alpha}\partial^{-i},\ \ \alpha=1,2,\cdots ,n,
\end{equation}
among them, $ R_{0\alpha}= E_{\alpha}$, $E_{\alpha}$ is a matrix having only one nonzero element on the $(\alpha,\alpha)$ place which is equal to $1$, $R_{\alpha}$ satisfies $[L, R_{\alpha}]=0$.\\
It can be found that $R_{j\alpha}$ exist. Their elements are differential polynomials composed of $u_{i}$, and they have the following properties
\begin{equation}
\ \ R_{\alpha}R_{\beta}=\delta_{\alpha\beta}R_{\alpha},\ \ \ \ \ \ \ \ \ \ \sum\limits_{\alpha=1}^{n}R_{\alpha}=I.
\end{equation}
$L$ and $R_{\alpha}$ are collectively called the Lax operators of the mcGD hierarchy.\\

The first definition of the mcGD hierarchy are the Lax equations
\begin{equation*}
\ \ \partial_{n,\alpha}L=[B_{n,\alpha},L],\ \ \ \ \ \ \ \ \ \ \
\partial_{n,\alpha}R_{\beta}=[B_{n,\alpha},R_{\beta}],\ \ n\neq 0\ mod\ N,\
\end{equation*}
\begin{equation}
\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}L=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \
\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}R_{\beta}=0,\ \ j=1,2,\cdots,\ \ \ \ \ \
\end{equation}
where $\partial_{n,\alpha}=\frac{\partial}{\partial t_{n,\alpha}}$, and $B_{n,\alpha}$ refers to the differential part of the operator $L^{\frac{n}{N}}R_{\alpha}$,
\begin{equation*}
B_{n,\alpha}=(L^{\frac{n}{N}}R_{\alpha})_{+}.
\end{equation*}
The second definition of the mcGD hierarchy is the zero curvature equation
\begin{equation}
\partial_{n,\alpha}B_{m,\alpha}-\partial_{m,\alpha}B_{n,\alpha}+[B_{m,\alpha},B_{n,\alpha}]=0,\ \ n,m\neq 0\ mod\ N,\ \
\end{equation}
the zero curvature equation of the mcGD hierarchy can be derived from its Lax equations.\\
Note: The variables $x$ and $t_{n,\alpha}$ are not independent, but they are also not equivalent. They have the following relationship
\begin{equation*}
\partial=\sum\limits_{\alpha=1}^{n}a_{\alpha}^{-\frac{1}{N}}\partial_{1,\alpha}.
\end{equation*}
When $N=2$, we can derive the multi-component KdV hierarchy; when $N=3$, we can derive the multi-component Boussinesq hierarchy.

The Lax operators of the mcGD hierarchy can also be expressed in dressing form
\begin{equation}
L=\phi A\partial^{N}\phi^{-1},\ \ \ \ \ R_{\alpha}=\phi E_{\alpha}\phi^{-1}, \ \ \ \ \ \
\end{equation}
where the quasi-differential operator $\phi=\phi(A\partial^{N})=\sum\limits_{i=1}^{\infty}\alpha_{i}(A\partial^{N})^{-i}, \alpha_{0}=I$, $\phi$ is called dressing operator or wave operator.
\begin{proposition}
By dressing transformation, the vector field on $A_{u}$ can be converted to $A_{w}$
\begin{equation*}
\partial_{n,\alpha}\phi=-(L^{\frac{n}{N}}R_{\alpha})_{-}\phi,\ \ \ n\neq 0\ mod\ N,
\end{equation*}
\begin{equation}
\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}\phi=0,\ \ \ j=1,2,\cdots,\ \ \ \ \ \ \ \ \ \ \
\end{equation}
the above equations are called the Sato equations of the multi-component Gelfand-Dickey hierarchy.
\end{proposition}
The wave function of the mcGD hierarchy is discussed below.\\
Let's first introduce a series
\begin{equation}
\ \ \xi(t,z)=\sum\limits_{i=1}^{\infty}\sum\limits_{\alpha=1}^{n}t_{i,\alpha}E_{\alpha}z^{i}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
$\xi(t,z)$ has the following properties
\begin{equation}\label{3.4}
\begin{split}
\partial_{n,\alpha}e^{\xi(t,z)}=z^{n}E_{\alpha}e^{\xi(t,z)}, \ \ n\neq 0\ mod\ N, \ \ \ \ \ \\
\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}e^{\xi(t,z)}=z^{jN}e^{\xi(t,z)},\ \ j=1,2,\cdots,\ \ \ \\                     \partial^{m}e^{\xi(t,z)}=z^{m}A^{-\frac{m}{N}}e^{\xi(t,z)}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{split}
\end{equation}
Then the wave function of the mcGD hierarchy can be defined as
\begin{equation*}
W(t,z)=\phi e^{\xi(t,z)}=\omega(t,z)e^{\xi(t,z)}.
\end{equation*}
\begin{Corollary}
The wave function of the multi-component Gelfand-Dickey hierarchy satisfies
\begin{equation}
\begin{split}
L^{\frac{m}{N}}W=z^{m}W,\ \ \ \ \ \ \ \ \partial_{n,\alpha}W=(L^{\frac{n}{N}}R_{\alpha})_{+}W, \ \ n\neq 0\ mod\ N,\ \ \ \ \ \ \ \ \\
\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}W=L^{j}W,\ \ j=1,2,\cdots, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{split}
\end{equation}
\end{Corollary}
\begin{proof}
According to the formula \eqref{3.4}, we can obtain
\begin{equation*}
A^{\frac{m}{N}}\partial^{m}e^{\xi(t,z)}=z^{m}e^{\xi(t,z)},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
so we can get
\begin{equation*}
A^{\frac{m}{N}}\partial^{m}=z^{m}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
therefore
\begin{equation*}
 \ \ \ L^{\frac{m}{N}}W=\phi A^{\frac{m}{N}}\partial^{m}e^{\xi(t,z)}=A^{\frac{m}{N}}\partial^{m}\phi e^{\xi(t,z)}=z^{m}W,
\end{equation*}
\begin{equation*}
\begin{split}
\ \ \ \ \ \ \ \ \ \ \partial_{n,\alpha}W=&\partial_{n,\alpha}(\phi e^{\xi(t,z)})=(\partial_{n,\alpha}\phi)e^{\xi(t,z)}+\phi(\partial_{n,\alpha}e^{\xi(t,z)})\\
=&-(L^{\frac{n}{N}}R_{\alpha})_{-}\phi e^{\xi(t,z)}+\phi z^{n}E_{\alpha}e^{\xi(t,z)}\\
=&-(L^{\frac{n}{N}}R_{\alpha})_{-}W+\phi A^{\frac{n}{N}}\partial^{n}E_{\alpha}\phi^{-1}\phi e^{\xi(t,z)}\\
=&-(L^{\frac{n}{N}}R_{\alpha})_{-}W+L^{\frac{n}{N}}R_{\alpha}W\\
=&(L^{\frac{n}{N}}R_{\alpha})_{+}W.
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}W=&\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}(\phi e^{\xi(t,z)})
=\phi(\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}e^{\xi(t,z)})\\ =&\phi(z^{jN}e^{\xi(t,z)})=L^{j}W.
\end{split}
\end{equation*}
\end{proof}
Next, let's consider the adjoint wave function $W^{*}(t,z)$, where the adjoint symbol $``\ast "$ represents a formal adjoint operator, such as $\partial^{*}=-\partial,\ (\partial^{-1})^{*}=-\partial^{-1},\ (AB)^{*}=B^{*}A^{*}$.\\
The adjoint wave function $W^{*}(t,z)$ of the mcGD hierarchy is defined as
\begin{equation}
W^{*}(t,z)=(\phi^{*})^{-1}e^{-\xi(t,z)}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
We have given the Lax operators of the mcGD hierarchy before, and it is easy to prove that the operators $\partial_{n,\alpha}-(L^{\frac{n}{N}}R_{\alpha})_{+}$ and $\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}-L^{j}$ can be expressed in a dressing form
\begin{equation*}
\partial_{n,\alpha}-(L^{\frac{n}{N}}R_{\alpha})_{+}=\phi(\partial_{n,\alpha}-A^{\frac{n}{N}}\partial^{n}E_{\alpha})\phi^{-1},\ \  n\neq 0\ mod\ N, \ \ \ \  \ \ \ \
\end{equation*}
\begin{equation}
\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}-L^{j}=\phi(\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}-A^{j}\partial^{jN})\phi^{-1},\ \ j=1,2,\cdots.\ \ \ \ \ \ \ \ \ \
\end{equation}
Dressing transformation on both sides of $[\partial_{n,\alpha}-A^{\frac{n}{N}}\partial^{n}E_{\alpha},A\partial^{N}]=0$ and $[\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}-A^{j}\partial^{jN},A\partial^{N}]=0$, then we can get the Lax equations of the mcGD hierarchy
\begin{equation*}
[\partial_{n,\alpha}-(L^{\frac{n}{N}}R_{\alpha})_{+},L]=0,\ \  n\neq 0\ mod\ N,\ \ \
\end{equation*}
\begin{equation}
[\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}-L^{j},L]=0,\ \  j=1,2,\cdots.\ \ \ \ \ \ \ \
\end{equation}

\subsection{Additional symmetry of the mcGD hierarchies}
\ \ \ \ We introduce an operator
\begin{equation}
\ \ \ \ \Gamma=\sum\limits_{k=1}^{\infty}\sum\limits_{\alpha=1}^{n}kt_{k,\alpha}A^{\frac{k-1}{N}}\partial^{k-1}E_{\alpha},\ \ k\neq 0\ mod\ N.
\end{equation}
The operator $\Gamma$ has the following properties
\begin{equation*}
\partial_{n,\alpha}\Gamma=nA^{\frac{n-1}{N}}\partial^{n-1}E_{\alpha},\ \ \
\partial^{k}\Gamma=\Gamma\partial^{k}+kA^{-\frac{1}{N}}\partial^{k-1}, \ \ \ \  n\neq 0\ mod\ N.
\end{equation*}
It is not difficult to verify that the operators $\Gamma$ and $\partial_{n,\alpha}-A^{\frac{n}{N}}\partial^{n}E_{\alpha}$ are commutative, that is to say,
\begin{equation}
[\partial_{n,\alpha}-A^{\frac{n}{N}}\partial^{n}E_{\alpha},\Gamma]=0, \ \ \  n\neq 0\ mod\ N.\ \
\end{equation}
Dressing transformation on both sides of $[\partial_{n,\alpha}-A^{\frac{n}{N}}\partial^{n}E_{\alpha},\Gamma]=0$, then we can obtain
\begin{equation}
\partial_{n,\alpha}M=[(L^{\frac{n}{N}}R_{\alpha})_{+},M], \ \ \  n\neq 0\ mod\ N,\ \
\end{equation}
where $M=\phi\Gamma\phi^{-1}$, we call it an Orlov-Shulman's operator.
\begin{definition}
The solution of the differential equation
\begin{equation}
\frac{\partial\phi}{\partial_{l,m,\alpha}^{*}}=-(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-}\phi, \ \ (\partial_{l,m,\alpha}^{*}=\frac{\partial}{\partial t_{l,m,\alpha}^{*}})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
is called the additional symmetry of the multi-component Gelfand-Dickey hierarchy.
\end{definition}
Next we consider a case related to the restriction of Virasoro. For the special differential operator $M^{m}L^{\frac{l}{N}}R_{\alpha}$, assuming its negative part disappears and let the operator $(M^{m}L^{\frac{l}{N}}R_{\alpha})_{+}$ act on $W$, we can get an equation related to $z$
\begin{equation}
(M^{m}L^{\frac{l}{N}}R_{\alpha})_{+}W=z^{l}E_{\alpha}\partial^{m}_{z}W.
\end{equation}
Note:This system can be rewritten into a linear equation for the isomonodromy problem.
\begin{Corollary}
Combining with the definition of the additional symmetry of the mcGD hierarchy, we can obtain
\begin{equation*}
\partial_{l,m,\alpha}^{*}L=-[(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-},L], \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
\begin{equation}
\partial_{l,m,\alpha}^{*}R_{\beta}=-[(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-},R_{\beta}], \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
these imply
\begin{equation}
\ \ \ \ \ \ \partial_{l,m,\alpha}^{*}(L^{\frac{n}{N}}R_{\beta})_{-}=-[(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-},(L^{\frac{n}{N}}R_{\beta})_{-}+\partial_{n,\beta}]_{-},\ \  n\neq 0\ mod\ N.
\end{equation}
\end{Corollary}
\begin{proof}
By calculating, we can get
\begin{equation*}
\begin{split}
&\partial_{l,m,\alpha}^{*}L\\
=&\partial_{l,m,\alpha}^{*}(\phi A\partial^{N}\phi^{-1})\\
=&(\partial_{l,m,\alpha}^{*}\phi)A\partial^{N}\phi^{-1}+\phi A\partial^{N}(\partial_{l,m,\alpha}^{*}\phi^{-1})\\
=&-(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-}L+L(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-}\\
=&-[(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-},L],\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{split}
\end{equation*}
The second equation can be proved by the same principle. Then we can find
\begin{equation*}
\begin{split}
&\partial_{l,m,\alpha}^{*}(L^{\frac{n}{N}}R_{\beta})_{-}\\
=&-[(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-},(L^{\frac{n}{N}}R_{\beta})_{-}]_{-}-[(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-},\partial_{n,\beta}]_{-}\\
=&-[(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-},(L^{\frac{n}{N}}R_{\beta})_{-}+\partial_{n,\beta}]_{-}. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{split}
\end{equation*}
\end{proof}
For the additional flows of the mcGD hierarchy, only a few special additional flows can survive.
\begin{theorem}\label{theorem-1}
In the additional flows of the multi-component Gelfand-Dickey hierarchy, only the flows which satisfy the condition $(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0$ and are shaped like  $\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*}$ can survive.
\end{theorem}
\begin{proof}
For the Lax operator $L$ of the mcGD hierarchy, it has no negative part, that is to say, its negative part is equal to zero.\\
From above, we can know $\partial_{l,m,\alpha}^{*}L=-[(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-},L]$, so let's consider
\begin{equation*}
\partial_{l,m,\alpha}^{*}L_{-}=-[(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-},L]_{-}=-(\phi[\Gamma^{m}A^{\frac{l}{N}}\partial^{l}E_{\alpha},A\partial^{N}]\phi^{-1})_{-},
\end{equation*}
after some deductions, the upper formula can be reduced to
\begin{equation*}
\partial_{l,m,\alpha}^{*}L_{-}=\frac{N}{N+l}\big[L^{\frac{N+l}{N}}R_{\alpha},M^{m}\big]_{-}=mN(M^{m-1}L^{\frac{N+l-1}{N}}R_{\alpha})_{-},\ \ \ \
\end{equation*}
if and only if $(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0$, we can get
\begin{equation*}
\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*}L_{-}=mN(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0.
\end{equation*}
\end{proof}
So we just need to thing about the surviving additional flows $\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*}$ which satisfy the condition $(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0$.\\
\ \ \ \ According to the theorem \ref{theorem-1}, we can find that assuming that a solution of the mcGD hierarchy is defined by the Virasoro condition $\sum\limits_{\alpha=1}^{n}(M^{m-1}L^{\frac{N+l-1}{N}}R_{\alpha})_{-}=0$, if the constraint $(L^{\frac{N+l}{N}})_{-}=0$ is imposed on this solution, it must also satisfy the $W_{1+\infty}$ symmetry $(M^{m})_{-}=0$. That is, the $W_{1+\infty}$ symmetry is compatible both with the Virasoro condition $(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0$ and the constraint $(L^{\frac{N+l}{N}})_{-}=0$.
\begin{proposition}\label{2.7}
The additional flows $\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*}$ which satisfy the condition $(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0$ commute with the flows $\partial_{n,\beta}\ (n\neq 0\ mod\ N)\ $ of the multi-component Gelfand-Dickey hierarchy.
\end{proposition}
\begin{proof}
\begin{equation*}
\ \ \ \ \ \ \ \ \big[\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*},\partial_{n,\beta}\big]\phi=\Big(\sum\limits_{\alpha=1}^{n}\big[\partial_{l,m,\alpha}^{*},\partial_{n,\beta}\big]\Big)\phi=
\sum\limits_{\alpha=1}^{n}\Big(\big[\partial_{l,m,\alpha}^{*},\partial_{n,\beta}\big]\phi\Big),
\end{equation*}
\begin{equation*}
\begin{split}
&[\partial_{l,m,\alpha}^{*},\partial_{n,\beta}]\phi\\
=&\partial_{l,m,\alpha}^{*}(\partial_{n,\beta}\phi)-\partial_{n,\beta}(\partial_{l,m,\alpha}^{*}\phi)\\
=&-\partial_{l,m,\alpha}^{*}((L^{\frac{n}{N}}R_{\beta})_{-}\phi)+\partial_{n,\beta}((M^{m}L^{\frac{l}{N}}R_{\alpha})_{-}\phi)\\
=&-(\partial_{l,m,\alpha}^{*}(L^{\frac{n}{N}}R_{\beta})_{-})\phi-(L^{\frac{n}{N}}R_{\beta})_{-}(\partial_{l,m,\alpha}^{*}\phi)\\
&+(\partial_{n,\beta}(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-})\phi+(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-}(\partial_{n,\beta}\phi)\\
=&[(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-},(L^{\frac{n}{N}}R_{\beta})_{-}+\partial_{n,\beta}]_{-}\phi+(L^{\frac{n}{N}}R_{\beta})_{-}(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-}\phi\\
&+[\partial_{n,\beta},(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-}]_{-}\phi-(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-}(L^{\frac{n}{N}}R_{\beta})_{-}\phi\\
=&[(L^{\frac{n}{N}}R_{\beta})_{-},(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-}]\phi-[(L^{\frac{n}{N}}R_{\beta})_{-},(M^{m}L^{\frac{l}{N}}R_{\alpha})_{-}]_{-}\phi\\
=&0,
\end{split}
\end{equation*}
then we can obtain $\big[\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*},\partial_{n,\beta}\big]\phi=0$.\\
From the above calculation we can find they commute on $\phi$, so they will commute on the whole differential algebra generated by coefficients of $\phi$. The proposition is proved.
\end{proof}
A direct application of the additional symmetry of the mcGD hierarchy is to derive the string equation which appears in the study of the string theory. First of all, we found the following relationships between the Lax operators and the Orlov-Shulman's operator of the mcGD hierarchy\\[5pt]
$[L^{\frac{1}{N}},M]=\phi[A^{\frac{1}{N}}\partial,\Gamma]\phi^{-1}=\phi(A^{\frac{1}{N}}\partial\Gamma)\phi^{-1}=\phi(A^{\frac{1}{N}}A^{-{\frac{1}{N}}})\phi^{-1}=I$,\\[5pt]
$[R_{\alpha},M]=\phi[E_{\alpha},\Gamma]\phi^{-1}=O.$\\[5pt]
Then, we can get
\begin{equation}\label{o}
[L^{\frac{n}{N}},M]=\phi[A^{\frac{n}{N}}\partial^{n},\Gamma]\phi^{-1}=\phi(A^\frac{n}{N}nA^{-\frac{1}{N}}\partial^{n-1})\phi^{-1}=nL^{\frac{n-1}{N}}.
\end{equation}
Furthermore, we can obtain
$[L^{\frac{n}{N}},ML^{-\frac{n-1}{N}}R_{\alpha}]=nR_{\alpha}.$\\
Next, we consider a special additional flow $\big(l=-(n-1),\ l-1=0\ mod\ N\big)$
\begin{equation*}
\sum\limits_{\alpha=1}^{n}\partial_{-(n-1),1,\alpha}^{*}L^{\frac{n}{N}}=-\sum\limits_{\alpha=1}^{n}\big[(ML^{-\frac{n-1}{N}}R_{\alpha})_{-},L^{\frac{n}{N}}\big],\ \ n=0\ mod\ N.
\end{equation*}
Combined with the formula \eqref{o}, we can rewrite it to
\begin{equation*}
\sum\limits_{\alpha=1}^{n}\partial_{-(n-1),1,\alpha}^{*}L^{\frac{n}{N}}=\sum\limits_{\alpha=1}^{n}\big[(ML^{-\frac{n-1}{N}}R_{\alpha})_{+},L^{\frac{n}{N}}\big]+nI, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
and when $n=0\ mod\ N$, there is $L^{\frac{n}{N}}=(L^{\frac{n}{N}})_{+},$
therefore
\begin{equation*}
\sum\limits_{\alpha=1}^{n}\big[L^{\frac{n}{N}},\frac{1}{n}(ML^{-\frac{n-1}{N}}R_{\alpha})_{+}\big]=I. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
so
\begin{equation}\label{p}
\Big[L^{\frac{n}{N}},\frac{1}{n}\big(\sum\limits_{\alpha=1}^{n}(ML^{-\frac{n-1}{N}}R_{\alpha})_{+}\big)\Big]=I. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
The equation \eqref{p} is called the string equation of the mcGD hierarchy.
From the above deduction process, we can find that the string equation refers to the condition that the operator is independent of the additional variables.\\

\subsection{$\tau$ function and Virasoro constraint of the mcGD hierarchies}
\ \ \ \ The existence theorem of the $\tau$ function is given below.
\begin{theorem}\cite{20}
Suppose the $\tau$ function is a matrix $\tau=(\tau_{\alpha,\beta})$ and $\tau_{\alpha,\alpha}$ is abbreviated as $\tau$, then there exists a function $\tau(\cdots,t_{s,\gamma},\cdots)$, which makes
\begin{equation}
\begin{cases}
\omega_{\alpha,\alpha}(t,z)=\frac{\tau(\cdots,t_{s,\gamma}-\delta_{\gamma,\beta}\frac{1}{sz^{s}},\cdots)}{\tau(\cdots,t_{s,\gamma},\cdots)},\\[5pt]
\omega_{\alpha,\beta}(t,z)=\frac{\tau_{\alpha,\beta}(\cdots,t_{s,\gamma}-\delta_{\gamma,\beta}\frac{1}{sz^{s}},\cdots)}{z\cdot\tau(\cdots,t_{s,\gamma},\cdots)},\ \alpha\neq\beta,\ \ s\neq 0\ mod\ N,
\end{cases}
\end{equation}
only when $\gamma=\beta$, the time variable $t_{s,\gamma}$ will be moved.\\
 Except for the above case $s\neq 0\ mod\ N$, the other cases all satisfy
\begin{equation}
\sum\limits_{\alpha=1}^{n}\partial_{kN,\alpha}\tau=0,\ \ k=1,2,\cdots.
\end{equation}
\end{theorem}
Next, starting from the additional symmetry of the wave operator $\phi$ and combining with the existence theorem of the $\tau$ function, we study the additional symmetry of the $\tau$ function.
To facilitate subsequent calculations, we first split $\sum\limits_{\alpha=1}^{n}\partial_{l,1,\alpha}^{*}\phi\ (l-1=0\ mod\ N)$ appropriately, where
\begin{equation*}
\begin{split}
\sum\limits_{\alpha=1}^{n}\partial_{l,1,\alpha}^{*}\phi=-\sum\limits_{\alpha=1}^{n}(ML^{\frac{l}{N}}R_{\alpha})_{-}\phi=-\sum\limits_{\alpha=1}^{n}(\phi\sum\limits_{k=1}^{\infty}
kt_{k,\alpha}A^{\frac{k+l-1}{N}}\partial^{k+l-1}E_{\alpha}\phi^{-1})_{-}\phi,\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k\neq 0\ mod\ N,
\end{split}
\end{equation*}
based on the value of $k$, we can divide $(\phi\sum\limits_{k=1}^{\infty}
kt_{k,\alpha}A^{\frac{k+l-1}{N}}\partial^{k+l-1}E_{\alpha}\phi^{-1})_{-}\phi\ (l<0)$ of the above formula into three parts
\begin{equation*}
O=[\phi,t_{1,\alpha}]A^{\frac{l}{N}}\partial^{l}E_{\alpha}+\sum\limits_{k=1}^{-l}
kt_{k,\alpha}\phi A^{\frac{k+l-1}{N}}\partial^{k+l-1}E_{\alpha},
\end{equation*}
\begin{equation*}
P=(\phi(-l+1)t_{-l+1,\alpha}E_{\alpha}\phi^{-1})_{-}\phi=(-l+1)[\phi,t_{-l+1,\alpha}E_{\alpha}],
\end{equation*}
\begin{equation*}
Q=-\sum\limits_{k=-l+2}^{\infty}kt_{k,\alpha}\partial_{k+l-1,\alpha}\phi,
\end{equation*}
from the above splitting results, we can get
\begin{equation}
\begin{split}
\sum\limits_{\alpha=1}^{n}\partial_{l,1,\alpha}^{*}\omega=&-z^{l}\partial_{z}\omega I-\sum\limits_{\alpha=1}^{n}\big(\sum\limits_{k=1}^{-l}kt_{k,\alpha}z^{k+l-1}\omega E_{\alpha}+(-l+1)[\omega,t_{-l+1,\alpha}E_{\alpha}]\ \ \\
&-\sum\limits_{k=-l+2}^{\infty}kt_{k,\alpha}\partial_{k+l-1,\alpha}\omega\big), \ \ \ k\neq 0\ mod\ N,\ \ \ \ \ \ \ \ \ \ \ \
\end{split}
\end{equation}
accordingly, we can obtain
\begin{equation}
\begin{split} \ \ \sum\limits_{\alpha=1}^{n}\partial_{l,1,\alpha}^{*}\omega_{\gamma,\beta}=&\sum\limits_{\alpha=1}^{n}\big(-z^{l}\partial_{z}\omega_{\gamma,\beta}\delta_{\alpha,\beta}-\sum\limits_{k=1}^{-l}kt_{k,\alpha}z^{k+l-1}\omega_{\gamma,\beta}\delta_{\alpha,\beta}-(-l+1)[\omega,t_{-l+1,\alpha}\delta_{\alpha,\beta}]\\
&+\sum\limits_{k=-l+2}^{\infty}kt_{k,\alpha}\partial_{k+l-1,\alpha}\omega_{\gamma,\beta}\big),\ \ \ k\neq 0\ mod\ N.\ \ \ \
\end{split}
\end{equation}
After the above study of relevant knowledge, it is easy to obtain the proposition on the additional symmetry of the $\tau$ function.
\begin{proposition}\label{r}\cite{20}
The action of additional flows $\sum\limits_{\alpha=1}^{n}\partial_{l,1,\alpha}^{*}\ (l-1=0\ mod\ N)$ with $l<0$ on $(\tau_{\gamma,\beta})$ is given by
\begin{equation}
\begin{split}
\sum\limits_{\alpha=1}^{n}\partial_{l,1,\alpha}^{*}\tau_{\gamma,\beta}=\sum\limits_{\alpha=1}^{n}(\sum\limits_{k=-l+1}^{\infty}kt_{k,\alpha}\partial_{k+l-1,\alpha}+\frac{1}{2}
&\sum\limits_{k+s=-l+1}kst_{k,\alpha}t_{s,\alpha})\tau_{\gamma,\beta}+(\sum\limits_{\alpha=1}^{n}c_{\alpha,\beta})\tau_{\gamma,\beta}.
\end{split}
\end{equation}
where $t_{k,\alpha},\ t_{s,\alpha}$ are arguments of the element $\tau_{\gamma,\beta}$.
\end{proposition}
The method of proving the above proposition is similar to that in the literature\cite{20}, and it will not be repeated here.
Because the string equation is the condition that the operator is independent of the additional variables, we can also obtain the condition that the operators are independent of the additional variables by the way of proving the above proposition, as shown below,
\begin{equation}
L_{l}\tau=0,\ \ l-1=0\ mod\ N,\ l<0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
where
\begin{equation*}
\begin{split}
L_{l}=\sum\limits_{\alpha=1}^{n}\big(\sum\limits_{k=-l+1}^{\infty}kt_{k,\alpha}&\partial_{k+l-1,\alpha}+\frac{1}{2}\sum\limits_{k+s=-l+1}kst_{k,\alpha}t_{s,\alpha}+c_{\alpha,\beta}\big),
\end{split}
\end{equation*}
it is obviously equivalent to the string equation, We call it the Virasoro constraint of the mcGD hierarchy. And $L_{l}$ satisfies the Virasoro exchange relation
\begin{equation*}
[L_{-m},L_{-n}]=(-m+n)L_{-(m+n)},\ \ m,n=1,2,\cdots.\ \ \ \
\end{equation*}
There are non-autonomous ODEs, related to the solutions of the Virasoro constraints, similar to the Painlev$\acute{e}$ equations appearing in the theory of equations NLS and KdV. Next, we'll give a brief description based on the classical KdV equation.
\begin{example}\cite{23}
The classical KdV equation
\begin{equation}\label{g}
u_{t}+6uu_{x}+u_{xxx}=0
\end{equation}
can be obtained by Lax pairs $L=A\partial^{2}+u_{0}$ and $\mathbb{A}=-4\partial^{3}-3u\partial-3u_{x}$, where $A=diag(1),\ u_{0}=diag(u(x,t))$.
Now we introduce a Virasoro operator
\begin{equation*}
\hat{L}=L_{-1}=\frac{t_{1,1}^{2}}{2}+\sum\limits_{k=2}^{\infty}kt_{k,\alpha}\partial_{k-2,\alpha}.
\end{equation*}
Let $\tau(\mathbf{x})$ be a solution of the mcGD hierarchy, which satisfies $\hat{L}\tau(\mathbf{x})=0,$ add the limit $\frac{\partial\tau}{\partial t_{2i,1}}=0,\ (i=1,2,\cdots)$ to $\tau(\mathbf{x})$ to make it also a solution of the mcKdV hierarchy. For example, $\tau(\mathbf{x})$ satisfies the bilinear differential equation
\begin{equation}
(D_{t_{1,1}}^{4}+D_{t_{1,1}}D_{t_{3,1}})\tau\cdot\tau=0
\end{equation}
of equation\eqref{g}
Then let $\delta(\mathbf{x})=\tau(\mathbf{x})$ under the constraints $t_{1,1}=x,\ t_{5,1}=-\frac{1}{5},\ and\ t_{j,1}=0,\ (j\neq 1,5)$, it can be found that $\delta(\mathbf{x})$ satisfies the bilinear differential equation
\begin{equation}\label{m1}
(D_{x}^{4}-2x)\delta\cdot\delta=0.
\end{equation}
We take $p=-\frac{d^{2}log(\delta)}{dx^{2}}$, then equation\eqref{m1} can be rewritten into the first Painlev$\acute{e}$ equation $P_{\uppercase\expandafter{\romannumeral1}}$
\begin{equation*}
\frac{d^{2}p}{dx^{2}}=6p^{2}+x.
\end{equation*}
\end{example}

\section{B type multi-component Gelfand-Dickey hierarchies}
\ \ \ \ The second part of this paper has given the Lax operators of the mcGD hierarcy
\begin{equation*}
\ \ \ \ \ \ \ \ \ L=A\partial^{N}+u_{2}\partial^{N-2}+u_{3}\partial^{N-3}+\cdots +u_{0},
\end{equation*}
\begin{equation*}
R_{\alpha}=\sum\limits_{i=1}^{\infty}R_{i\alpha}\partial^{-i},\ \ \alpha=1,2,\cdots ,n,
\end{equation*}
if $L$ and $R_{\alpha}$ also satisfy $L^{*}=-\partial L\partial^{-1},\ R_{\alpha}^{*}=\partial R_{\alpha}\partial^{-1}$, then $L$ and $R_{\alpha}$ at this time are called the Lax operators of the mcBGD hierarchy.\\
From the above Lax operators, we can define the Lax equations of the mcBGD hierarchy
\begin{equation*}
\partial_{n,\alpha}L=[B_{n,\alpha},L],\\ \ \ \ \ \ \
\partial_{n,\alpha}R_{\beta}=[B_{n,\alpha},R_{\beta}],\ \ n\neq 0\ mod\ N,\ \ \ \ \ \ \
\end{equation*}
\begin{equation}
\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}L=0,\ \ \ \ \ \ \ \ \
\sum\limits_{\alpha=1}^{n}\partial_{jN,\alpha}R_{\beta}=0,\ \ j=1,2,\cdots,\ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
where $L^{*}=-\partial L\partial^{-1},\ R_{\alpha}^{*}=\partial R_{\alpha}\partial^{-1}$, $\partial_{n,\alpha}=\frac{\partial}{\partial t_{n,\alpha}}$ and $B_{n,\alpha}=(L^{\frac{n}{N}}R_{\alpha})_{+}$.
\begin{proposition}
The B type multi-component Gelfand-Dickey hierarchy has only odd flows.
\end{proposition}
\begin{proof}
From the foregoing, we can see that the Lax equations of the mcBGD hierarchy are
\begin{equation*}
\partial_{n,\alpha}L=[B_{n,\alpha},L],\\ \ \ \ \ \ \
\partial_{n,\alpha}R_{\beta}=[B_{n,\alpha},R_{\beta}],\ \ n\neq 0\ mod\ N,\ \ \ \ \ \ \
\end{equation*}
where $L^{*}=-\partial L\partial^{-1},\ R_{\alpha}^{*}=\partial R_{\alpha}\partial^{-1}$, $\partial_{n,\alpha}=\frac{\partial}{\partial t_{n,\alpha}}$ and $B_{n,\alpha}=(L^{\frac{n}{N}}R_{\alpha})_{+}$.\\
Combined with the constraints of its Lax equations, we can get
\begin{equation*}
(L^{\frac{n}{N}})^{*}=(L^{*})^{\frac{n}{N}}=(-1)^{n}\partial L^{\frac{n}{N}}\partial^{-1},
\end{equation*}
\begin{equation*}
\ \ \ \ \ \ \ \ \ \ \ \ (L^{\frac{n}{N}}R_{\alpha})^{*}=R_{\alpha}^{*}(L^{*})^{\frac{n}{N}}=(-1)^{n}\partial R_{\alpha}L^{\frac{n}{N}}\partial^{-1}.
\end{equation*}
Then we simplify $\partial_{n,\alpha}L^{*}$ and $\partial_{n,\alpha}R_{\alpha}^{*}$ in two ways
\begin{equation}\label{a}
\begin{split}
\partial_{n,\alpha}L^{*}=(\partial_{n,\alpha}L)^{*}=[B_{n,\alpha},L]^{*}=-\partial [B_{n,\alpha},L]\partial^{-1}=[\partial B_{n,\alpha}\partial^{-1},L^{*}],\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\partial_{n,\alpha}L^{*}=[B_{n,\alpha},L]^{*}=(B_{n,\alpha}L-L B_{n,\alpha})^{*}=L^{*}B_{n,\alpha}^{*}-B_{n,\alpha}^{*}L^{*}=[-B_{n,\alpha}^{*},L^{*}],\ \ \ \ \ \ \ \ \ \ \ \ \ \\[-15pt]
\end{split}
\end{equation}
\begin{equation}\label{b}
\begin{split}
\partial_{n,\alpha}R_{\beta}^{*}=(\partial_{n,\alpha}R_{\beta})^{*}=[B_{n,\alpha},R_{\beta}]^{*}=\partial [B_{n,\alpha},R_{\beta}]\partial^{-1}=[\partial B_{n,\alpha}\partial^{-1},R_{\beta}^{*}],\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\partial_{n,\alpha}R_{\beta}^{*}=[B_{n,\alpha},R_{\beta}]^{*}=(B_{n,\alpha}R_{\beta}-R_{\beta} B_{n,\alpha})^{*}=R_{\beta}^{*}B_{n,\alpha}^{*}-B_{n,\alpha}^{*}R_{\beta}^{*}=[-B_{n,\alpha}^{*},R_{\beta}^{*}],
\end{split}
\end{equation}
comparing equation \eqref{a} and \eqref{b}, we can find
\begin{equation}\label{c}
\ \ \ \ \ B_{n,\alpha}^{*}=-\partial B_{n,\alpha}\partial^{-1},\ \ n\neq 0\ mod\ N.
\end{equation}
Then we start with the definition of $B_{n,\alpha}\ (n\neq 0\ mod\ N)$ and solve its adjoint
\begin{equation}\label{d}
B_{n,\alpha}^{*}=((L^{\frac{n}{N}}R_{\alpha})_{+})^{*}=(-1)^{n}\partial (R_{\alpha}L^{\frac{n}{N}})_{+}\partial^{-1}=(-1)^{n}\partial B_{n,\alpha}\partial^{-1},\ \ n\neq 0\ mod\ N.\
\end{equation}
By analyzing equation \eqref{c} and \eqref{d}, we can find that $n$ can only take odd numbers.
\end{proof}
Therefore, a more concise form of the Lax equations of the mcBGD hierarchy can be obtained
\begin{equation*}
\partial_{2n+1,\alpha}L=[B_{2n+1,\alpha},L],\\ \ \ \ \ \partial_{2n+1,\alpha}R_{\beta}=[B_{2n+1,\alpha},R_{\beta}],\ \ 2n+1\neq 0\ mod\ N,
\end{equation*}
\begin{equation}
\sum\limits_{\alpha=1}^{n}\partial_{(2j^{'}+1)N,\alpha}L=0,\ \ \ \ \ \ \ \
\sum\limits_{\alpha=1}^{n}\partial_{(2j^{'}+1)N,\alpha}R_{\beta}=0,\ \ j^{'}=0,1,2,\cdots,\ \ \ \ \
\end{equation}
where $\partial_{2n+1,\alpha}=\frac{\partial}{\partial t_{2n+1,\alpha}}$ and $B_{2n+1,\alpha}=(L^{\frac{2n+1}{N}}R_{\alpha})_{+}$.\\
The Lax operators of the mcBGD hierarchy can also be expressed in dressing form
\begin{equation*}
L=\phi A\partial^{N}\phi^{-1},\ \ \ \ \ \ \ \ \ \ R_{\alpha}=\phi E_{\alpha}\phi^{-1},\ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
where dressing operator $\phi=\phi(A\partial^{N})=\sum\limits_{i=1}^{\infty}\alpha_{i}(A\partial^{N})^{-i},\ \alpha_{0}=I\ and\ \phi^{*}=\partial \phi^{-1}\partial^{-1}$
which is different from the mcGD hierarchy.\\
Subsequently, the Sato equations $\partial_{2n+1,\alpha}\phi=-(B_{2n+1,\alpha})_{-}\phi\ (2n+1\neq 0\ mod\ N)$ and $\sum\limits_{\alpha=1}^{n}\partial_{(2j^{'}+1)N,\alpha}\phi=0\ (j^{'}=0,1,2,\cdots)$ of the mcBGD hierarchy can also be obtained.\\
Then, the wave function $W(t,z)$ and the adjoint wave function $W^{*}(t,z)$ of the mcBGD hierarchy are given
\begin{equation*}
W(t,z)=\phi e^{\xi(t,z)}=\omega(t,z)e^{\xi(t,z)},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
\begin{equation}
W^{*}(t,z)=(\phi^{*})^{-1}e^{-\xi(t,z)}=(\partial\phi\partial^{-1})e^{-\xi(t,z)},\ \ \ \ \ \
\end{equation}
where
\begin{equation*}
\ \ \ \ \xi(t,z)=\sum\limits_{i=1}^{\infty}\sum\limits_{\alpha=1}^{n}t_{2i-1,\alpha}E_{\alpha}z^{2i-1}. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
It should be noted that the Lax equations of the mcBGD hierarchy can also be deduced from
\begin{equation*}
\begin{split}
L^{\frac{2n+1}{N}}W=z^{2n+1}W,\ \ \ \ \ \partial_{2n+1,\alpha}W=B_{2n+1,\alpha}W,\ \ \ 2n+1\neq 0\ mod\ N,\ \ \ \ \ \ \ \ \ \ \ \\
\sum\limits_{\alpha=1}^{n}\partial_{(2j^{'}+1)N,\alpha}W=L^{2j^{'}+1}W,\ \ j^{'}=0,1,2,\cdots. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{split}
\end{equation*}

\subsection{Additional symmetry of the mcBGD hierarchies}
\ \ \ \ Firstly, we give the  Orlov-Shulman's operator $M=\phi\Gamma\phi^{-1}$ of the mcBGD hierarchy, among them, $\Gamma=\sum\limits_{i=1}^{\infty}\sum\limits_{\alpha=1}^{n}(2i-1)t_{2i-1,\alpha}A^{\frac{2i-2}{N}}\partial^{2i-2}E_{\alpha},\ \ 2i-1\neq 0\ mod\ N.$\\
Combining the Orlov-Shulman's operator $M$, we can easily get $[M,L^{\frac{2n+1}{N}}]=-(2n+1)L^{\frac{2n}{N}}$.
\begin{definition}
The solution of the differential equation
\begin{equation}
\frac{\partial\phi}{\partial_{l,m,\alpha}^{*}}=-(D_{l,m,\alpha})_{-}\phi,
\end{equation}
where
\begin{equation*}
\ \ \partial_{l,m,\alpha}^{*}=\frac{\partial}{\partial t_{l,m,\alpha}^{*}},\ \ D_{l,m,\alpha}=M^{m}L^{\frac{l}{N}}R_{\alpha}-(-1)^{l}R_{\alpha}L^{\frac{l-1}{N}}M^{m}L^{\frac{1}{N}},
\end{equation*}
is called the additional symmetry of the B type multi-component Gelfand-Dickey hierarchy.
\end{definition}
Similarly, for the differential operator $D_{l,m,\alpha}$, assuming its negative part disappears and let the operator $(D_{l,m,\alpha})_{+}$ act on $W$, we can get an equation related to $z$
\begin{equation}
(D_{l,m,\alpha})_{+}W=(z^{l}-(-1)^{l}z^{2l-1})E_{\alpha}\partial^{m}_{z}W-(-1)^{l}m(l-1)z^{2(l-1)}E_{\alpha}\partial^{m-1}_{z}W.
\end{equation}
Note:This system can also be rewritten into a linear equation for the isomonodromy problem.\\
According to the definition of the additional symmetry of the mcBGD hierarchy, some formulas can be obtained by simple calculation
\begin{equation}
\partial_{l,m,\alpha}^{*}L=-[(D_{l,m,\alpha})_{-},L], \ \ \ \ \ \ \
\partial_{l,m,\alpha}^{*}R_{\beta}=-[(D_{l,m,\alpha})_{-},R_{\beta}].
\end{equation}
For the additional flows of the mcBGD hierarchy, they have a fraction of the flows that can survive.
\begin{theorem}
In the additional flows of the B type multi-component Gelfand-Dickey hierarchy, the flows which satisfy the condition $(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0$ and are shaped like $\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*}$ or the condition $l=2i\ (i\in\mathbb{Z})$ and are shaped like $\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*}$ can survive.
\end{theorem}
\begin{proof}
For Lax operator $L$ of the mcBGD hierarchy, its negative part is equal to zero.\\
And we know $\partial_{l,m,\alpha}^{*}L=-[(D_{l,m,\alpha})_{-},L]$, so let's consider
\begin{equation*}
\begin{split}
\ \ \ \ \partial_{l,m,\alpha}^{*}L_{-}=&-[(D_{l,m,\alpha})_{-},L]_{-}=-(\phi[\Gamma^{m}A^{\frac{l}{N}}\partial^{l}E_{\alpha},A\partial^{N}]\phi^{-1})_{-}\\
&+(-1)^{l}(\phi[E_{\alpha}A^{\frac{l-1}{N}}\partial^{l-1}\Gamma^{m}A^{\frac{1}{N}}\partial,A\partial^{N}]\phi^{-1})_{-},
\end{split}
\end{equation*}
after some deductions, the upper formula can be reduced to
\begin{equation*}
\partial_{l,m,\alpha}^{*}L_{-}=(1-(-1)^{l})mN(M^{m-1}L^{\frac{N+l-1}{N}}R_{\alpha})_{-},\ \ \ \ \ \ \ \
\end{equation*}
only when $(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0$ or $l=2i\ (i\in\mathbb{Z})$, we can obtain
\begin{equation*}
\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*}L_{-}=(1-(-1)^{l})mN(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0.\ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
\end{proof}
\begin{proposition}
The additional symmetric flows $\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*}$ which satisfy the condition $(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0$ or the condition $l=2i\ (i\in\mathbb{Z})$ commute with the flows $\partial_{2k+1,\beta}\ (2k+1\neq0\ mod\ N)$ of the B type multi-component Gelfand-Dickey hierarchy.
\end{proposition}
\begin{proof}
\begin{equation*}
\ \ \ \ \ \ \ \ \big[\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*},\partial_{2k+1,\beta}\big]\phi=\Big(\sum\limits_{\alpha=1}^{n}\big[\partial_{l,m,\alpha}^{*},\partial_{2k+1,\beta}\big]\Big)\phi=
\sum\limits_{\alpha=1}^{n}\Big(\big[\partial_{l,m,\alpha}^{*},\partial_{2k+1,\beta}\big]\phi\Big),
\end{equation*}
\begin{equation*}
\begin{split}
&[\partial_{l,m,\alpha}^{*},\partial_{2k+1,\beta}]\phi\\
=&-\partial_{l,m,\alpha}^{*}((B_{2k+1,\beta})_{-}\phi)+\partial_{2k+1,\beta}((D_{l,m,\alpha})_{-}\phi)\\
=&[(D_{l,m,\alpha})_{-},(B_{2k+1,\beta})_{-}+\partial_{2k+1,\beta}]_{-}\phi+(B_{2k+1,\beta})_{-}(D_{l,m,\alpha})_{-}\phi\\ \ \ \ \ \ \ \
&+[\partial_{2k+1,\beta},(D_{l,m,\alpha})_{-}]_{-}\phi-(D_{l,m,\alpha})_{-}(B_{2k+1,\beta})_{-}\phi\\
=&[(D_{l,m,\alpha})_{-},(B_{2k+1,\beta})_{-}]\phi-[(D_{l,m,\alpha})_{-},(B_{2k+1,\beta})_{-}]_{-}\phi\\
=&0,
\end{split}
\end{equation*}
then $\big[\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*},\partial_{2k+1,\beta}\big]\phi=0,$ so the proposition is proved.
\end{proof}
Then we consider a special additional flow $\big(l=-(2k-1),\ l-1=0\ mod\ N\big)$
\begin{equation*}
\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}L^{\frac{2k}{N}}=-\sum\limits_{\alpha=1}^{n}\big[(D_{-(2k-1),1,\alpha})_{-},L^{\frac{2k}{N}}\big],\ 2k=0\ mod\ N,
\end{equation*}
after a series of calculations, we can get
\begin{equation*}
\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}L^{\frac{2k}{N}}=\sum\limits_{\alpha=1}^{n}\big[(D_{-(2k-1),1,\alpha})_{+},L^{\frac{2k}{N}}\big]+4kI,\ \
\end{equation*}
When $2k=0\ mod\ N$, $L^{\frac{2k}{N}}$ is a differential operator, then
\begin{equation*}
\sum\limits_{\alpha=1}^{n}\big[L^{\frac{2k}{N}},\frac{1}{4k}(D_{-(2k-1),1,\alpha})_{+}\big]=I,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
thus $\Big[L^{\frac{2k}{N}},\frac{1}{4k}\big(\sum\limits_{\alpha=1}^{n}(D_{-(2k-1),1,\alpha})_{+}\big)\Big]=I\ $ is the string equation of the mcBGD hierarchy.

\subsection{$\tau$ function and Virasoro constraint of the mcBGD hierarchies}
\ \ \ \ The existence theorem of a $\tau$ function of the mcBGD hierarchy is given below.
\begin{theorem}\cite{20}
Suppose the $\tau$ function is a matrix T=($\tau_{\alpha,\beta}$) and $\tau_{\alpha,\alpha}$ is abbreviated as $\tau$, then there exists a function $\tau(\cdots,t_{2s-1,\gamma},\cdots)$, which makes
\begin{equation}\label{1}
\begin{cases}
\omega_{\alpha,\alpha}(t,z)=\frac{\tau(\cdots,t_{2s-1,\gamma}-\delta_{\gamma,\beta}\frac{1}{(2s-1)z^{2s-1}},\cdots)}{\tau(\cdots,t_{2s-1,\gamma},\cdots)},\\[5pt]
\omega_{\alpha,\beta}(t,z)=\frac{\tau_{\alpha,\beta}(\cdots,t_{2s-1,\gamma}-\delta_{\gamma,\beta}\frac{1}{(2s-1)z^{2s-1}},\cdots)}{z\cdot\tau(\cdots,t_{2s-1,\gamma},\cdots)},\ \alpha\neq\beta,\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2s-1\neq 0\ mod\ N,
\end{cases}
\end{equation}
only when $\gamma=\beta$, the time variable $t_{2s-1,\gamma}$ will be moved.\\
 Except for the above case $2s-1\neq 0\ mod\ N$, the other cases satisfy $\sum\limits_{\alpha=1}^{n}\partial_{k,\alpha}\tau=0,\ where\ \ k=0\ mod\ N\ or\ k=0\ mod\ 2.$
\end{theorem}
Next, starting from the additional symmetry of the wave operator $\omega$ and combining with the existence theorem of the $\tau$ function, we study the additional symmetry of the $\tau$ function.
\begin{proposition}\label{e}
$D_{-(2k-1),1,\alpha}\ (2k=0\ mod\ N,\ k>0)$ can be reduced to the following form
\begin{equation}
\begin{split}
&(D_{-(2k-1),1,\alpha})_{-}\\
=&2\phi(\sum\limits_{i=1}^{k}(2i-1)t_{2i-1,\alpha}A^{\frac{2i-2k-1}{N}}\partial^{2i-2k-1}E_{\alpha})\phi^{-1}\\
&+2\sum\limits_{i=k+1}^{\infty}(2i-1)t_{2i-1,\alpha}(L^{\frac{2i-2k-1}{N}}R_{\alpha})_{-}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
&-2kL^{-\frac{2k}{N}}R_{\alpha},\ \ 2i-1\neq 0\ mod\ N.
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
According to the additional symmetry of the mcBGD hierarchy, we will have
\begin{equation*}
\begin{split}
\ \ \ \ \ &(D_{-(2k-1),1,\alpha})_{-}\\
=&(ML^{-\frac{2k-1}{N}}R_{\alpha}-(-1)^{-(2k-1)}R_{\alpha}L^{\frac{l-1}{N}}M^{m}L^{\frac{1}{N}})_{-}\\
=&(ML^{-\frac{2k-1}{N}}R_{\alpha})_{-}+(R_{\alpha}L^{\frac{l-1}{N}}M^{m}L^{\frac{1}{N}})_{-}\\
=&2(\phi\Gamma A^{-\frac{2k-1}{N}}\partial^{-(2k-1)}E_{\alpha}\phi^{-1})_{-}-2k(\phi E_{\alpha} A^{-\frac{2k}{N}}\partial^{-2k}\phi^{-1})_{-}\\
=&2(\phi\sum\limits_{i=1}^{\infty}(2i-1)t_{2i-1,\alpha}A^{\frac{2i-2k-1}{N}}\partial^{2i-2k-1}E_{\alpha}\phi^{-1})_{-}-2kL^{-\frac{2k}{N}}R_{\alpha}\\
=&2\phi(\sum\limits_{i=1}^{k}(2i-1)t_{2i-1,\alpha}A^{\frac{2i-2k-1}{N}}\partial^{2i-2k-1}E_{\alpha})\phi^{-1}\\
&+2\sum\limits_{i=k+1}^{\infty}(2i-1)t_{2i-1,\alpha}(L^{-\frac{2i-2k-1}{N}}R_{\alpha})_{-}
-2kL^{-\frac{2k}{N}}R_{\alpha}.
\end{split}
\end{equation*}
\end{proof}
Based on the additional symmetry of the mcBGD hierarchy and the proposition \ref{e}, we can calculate
\begin{equation}\label{f}
\begin{split}
&\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}\phi\\
=&-\sum\limits_{\alpha=1}^{n}(D_{-(2k-1),1,\alpha})_{-}\phi\\
=&\sum\limits_{\alpha=1}^{n}\big(-2\phi\sum\limits_{i=1}^{k}(2i-1)t_{2i-1,\alpha}A^{\frac{2i-2k-1}{N}}\partial^{2i-2k-1}E_{\alpha}\\
&+2\sum\limits_{i=k+1}^{\infty}(2i-1)t_{2i-1,\alpha}(\partial_{2(i-k)-1,\alpha}\phi)\big)
+2k\phi A^{-\frac{2k}{N}}\partial^{-2k}I,\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ (2k=0\ mod\ N).
\end{split}
\end{equation}
Obviously, when both sides of equation \eqref{f} are simultaneously applied to a function $exp(A^{-\frac{1}{N}}xz)$, the equation is still valid. By further calculating with
\begin{equation*}
[\phi,t_{1,\alpha}]exp(A^{-\frac{1}{N}}xz)=(\partial_{z}\omega)exp(A^{-\frac{1}{N}}xz),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
\begin{equation*}
\phi\partial^{-l}exp(A^{-\frac{1}{N}}xz)=\phi z^{-l}exp(A^{-\frac{1}{N}}xz)=z^{-l}\omega(exp(A^{-\frac{1}{N}}xz)),\ \ \
\end{equation*}
we can get
\begin{equation*}
\begin{split}
&(\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}\omega)exp(A^{-\frac{1}{N}}xz)\\
=&(-\sum\limits_{\alpha=1}^{n}(D_{-(2k-1),1,\alpha})_{-}\phi)exp(A^{-\frac{1}{N}}xz)\\
=&-2z^{-2k+1}(\partial_{z}\omega)exp(A^{-\frac{1}{N}}xz)+2kz^{-2k}\omega(exp(A^{-\frac{1}{N}}xz))\\
&-2(\sum\limits_{i=1}^{k}\sum\limits_{\alpha=1}^{n}(2i-1)t_{2i-1,\alpha}A^{\frac{2i-2k-1}{N}}\partial^{2i-2k-1}E_{\alpha}\omega)exp(A^{-\frac{1}{N}}xz)\\
&+2\sum\limits_{i=k+1}^{\infty}\sum\limits_{\alpha=1}^{n}(2i-1)t_{2i-1,\alpha}(\partial_{2(i-k)-1,\alpha}\omega)exp(A^{-\frac{1}{N}}xz),
\end{split}
\end{equation*}
therefore
\begin{equation}
\begin{split}
&\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}\omega\\
=&-2z^{-2k+1}(\partial_{z}\omega)I+2kz^{-2k}\omega I\\
&-2(\sum\limits_{i=1}^{k}\sum\limits_{\alpha=1}^{n}(2i-1)t_{2i-1,\alpha}A^{\frac{2i-2k-1}{N}}\partial^{2i-2k-1}E_{\alpha}\omega)\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \\
&+2\sum\limits_{i=k+1}^{\infty}\sum\limits_{\alpha=1}^{n}(2i-1)t_{2i-1,\alpha}(\partial_{2(i-k)-1,\alpha}\omega),\\
\end{split}
\end{equation}
correspondingly, we can obtain
\begin{equation}
\begin{split}
&\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}\omega_{\gamma,\beta}\\
=&\sum\limits_{\alpha=1}^{n}\big(-2z^{-2k+1}\delta_{\alpha,\beta}(\partial_{z}\omega_{\gamma,\beta})+2kz^{-2k}\delta_{\alpha,\beta}\omega_{\gamma,\beta}\\
&-2(\sum\limits_{i=1}^{k}(2i-1)t_{2i-1,\alpha}A^{\frac{2i-2k-1}{N}}\partial^{2i-2k-1}\delta_{\alpha,\beta}\omega_{\gamma,\beta})\ \ \ \ \ \ \ \ \ \ \
\ \\
&+2\sum\limits_{i=k+1}^{\infty}(2i-1)t_{2i-1,\alpha}(\partial_{2(i-k)-1,\alpha}\omega_{\gamma,\beta})\big).\\
\end{split}
\end{equation}
We can find that $\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}\phi=0$ and $\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}\omega=0\ $ are equivalent. Thus, we can get the constraints of the tau function by equation \eqref{1}.\\
For $\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}\omega\ (2k=0\ mod\ N,\ k>0)$, it is difficult to find a unified simplified form, so in this paper we only discuss the case where $k$ is a positive integer.
\begin{proposition}
The action of additional symmetries $\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}\ (2k=0\ mod\ N,\ k>0)$ on $(\tau_{\gamma,\beta})$ is given by
\begin{equation}
\begin{split}
&\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}\tau_{\gamma,\beta}\\
=&\sum\limits_{\alpha=1}^{n}\big((\frac{1}{2}\sum\limits_{m=2k}^{\infty}(2m-1)t_{2m-1,\alpha}\partial_{2(m-k)-1,\alpha}\\
&+\frac{1}{8}\sum\limits_{m+s=2k}(2m-1)(2s-1)t_{2m-1,\alpha}t_{2s-1,\alpha})\big)\tau_{\gamma,\beta}+(\sum\limits_{\alpha=1}^{n}c_{\alpha,\beta})\tau_{\gamma,\beta}.
\end{split}
\end{equation}
where $t_{2m-1,\alpha},\ t_{2s-1,\alpha}$ are arguments of the element $\tau_{\gamma,\beta}$.
\end{proposition}
Taking
\begin{equation}
\begin{split}
L_{-(2k-1)}=&\sum\limits_{\alpha=1}^{n}\big(\frac{1}{2}\sum\limits_{m=2k}^{\infty}(2m-1)t_{2m-1,\alpha}\partial_{2(m-k)-1,\alpha}\ \ \ \ \\
&+\frac{1}{8}\sum\limits_{m+s=2k}(2m-1)(2s-1)t_{2m-1,\alpha}t_{2s-1,\alpha}+c_{\alpha,\beta}\big),
\end{split}
\end{equation}
where $2k=0\ mod\ N,\ k>0$.\\
We can verify that $L_{-(2k-1)}$ satisfies the Virasoro exchange relation
\begin{equation}
[L_{-m},L_{-n}]=(-m+n)L_{-(m+n)},\ \ m,n=1,2,\cdots.\ \ \ \ \ \
\end{equation}
so the Virasoro constraint of the $\tau$ function of the mcBGD hierarchy is
\begin{equation}
L_{-(2k-1)}\tau=0,\ \ 2k=0\ mod\ N,\ k>0.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}

\section{C type multi-component Gelfand-Dickey hierarchies}
\ \ \ \ The Lax operators of the mcCGD hierarchy are composed of the operators $L$ and $R_{\alpha}$ of the mcGD hierarchy and the constraint conditions $L^{*}=(-1)^{N}L,\ R_{\alpha}^{*}=R_{\alpha}$. It is clear that the mcCGD hierarchy can be divided into two types according to the parity of $N$. The odd mcCGD hierarchy can be constructed according to the differential operator
\begin{equation*}
\ \ \ \ \ \ \ \ L=A\partial^{N}+u_{2}\partial^{N-2}+u_{3}\partial^{N-3}+\cdots +u_{0},\ N\neq 0\ mod\ 2,
\end{equation*}
\begin{equation*}
R_{\alpha}=\sum\limits_{i=1}^{\infty}R_{i\alpha}\partial^{-i},\ \ \alpha=1,2,\cdots ,n,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
which satisfy $L*=-L$ and $R_{\alpha}^{*}=R_{\alpha}$.\\
The even mcCGD hierarchy can also be constructed according to the differential operator
\begin{equation*}
\ \ \ \ \ \ \ \ \ L=A\partial^{N}+u_{2}\partial^{N-2}+u_{3}\partial^{N-3}+\cdots +u_{0},\ N=0\ mod\ 2,
\end{equation*}
\begin{equation*}
R_{\alpha}=\sum\limits_{i=1}^{\infty}R_{i\alpha}\partial^{-i},\ \ \alpha=1,2,\cdots ,n,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
which satisfy $L*=L$ and $R_{\alpha}^{*}=R_{\alpha}$.\\
Though their constraints are different, their additional symmetric generators, surviving flows, string equation and so on are the same.

Similar to the Lax equations of the mcBGD hierarchy, the Lax equations of the mcCGD hierarchy does not have even flows, that is, $u_{i}=u_{i}(x;t_{1},t_{3},\cdots)$.
\begin{proposition}
The C type multi-component Gelfand-Dickey hierarchy has only odd flows.
\end{proposition}
\begin{proof}
From the foregoing, we already know the Lax equations of the mcCGD hierarchy. Combined with the constraints of its Lax equations, we can get
\begin{equation*}
(L^{\frac{n}{N}})^{*}=(-1)^{n}L^{\frac{n}{N}},\ \ \ \ \
\end{equation*}
\begin{equation*}
\ \ \ (L^{\frac{n}{N}}R_{\alpha})^{*}=(-1)^{n}R_{\alpha}L^{\frac{n}{N}}.
\end{equation*}
Then we simplify $\partial_{n,\alpha}L^{*}$ and $\partial_{n,\alpha}R_{\alpha}^{*}$ in two ways
\begin{equation}\label{a1}
\begin{split}
\partial_{n,\alpha}L^{*}=(\partial_{n,\alpha}L)^{*}=[B_{n,\alpha},L]^{*}=-[B_{n,\alpha},L]=[B_{n,\alpha},L^{*}],\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\partial_{n,\alpha}L^{*}=[B_{n,\alpha},L]^{*}=(B_{n,\alpha}L-L B_{n,\alpha})^{*}=L^{*}B_{n,\alpha}^{*}-B_{n,\alpha}^{*}L^{*}=[-B_{n,\alpha}^{*},L^{*}],\ \ \ \ \ \ \ \\[-5pt]
\end{split}
\end{equation}
\begin{equation}\label{b1}
\begin{split}
\partial_{n,\alpha}R_{\beta}^{*}=(\partial_{n,\alpha}R_{\beta})^{*}=[B_{n,\alpha},R_{\beta}]^{*}=[B_{n,\alpha},R_{\beta}]=[B_{n,\alpha},R_{\beta}^{*}],\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\partial_{n,\alpha}R_{\beta}^{*}=[B_{n,\alpha},R_{\beta}]^{*}=(B_{n,\alpha}R_{\beta}-R_{\beta} B_{n,\alpha})^{*}=R_{\beta}^{*}B_{n,\alpha}^{*}-B_{n,\alpha}^{*}R_{\beta}^{*}=[-B_{n,\alpha}^{*},R_{\beta}^{*}],\ \ \ \ \ \
\end{split}
\end{equation}
comparing \eqref{a1} and \eqref{b1}, we can find
\begin{equation}\label{c1}
\ B_{n,\alpha}^{*}=-B_{n,\alpha},\ \ n\neq 0\ mod\ N.
\end{equation}
Then we combine the definition of $B_{n,\alpha}\ (n\neq 0\ mod\ N)$ to solve its adjoint
\begin{equation}\label{d1}
B_{n,\alpha}^{*}=((L^{\frac{n}{N}}R_{\alpha})_{+})^{*}=(-1)^{n}(R_{\alpha}L^{\frac{n}{N}})_{+}=(-1)^{n}B_{n,\alpha},\ \ n\neq 0\ mod\ N.\ \
\end{equation}
By analyzing equation \eqref{c1} and \eqref{d1}, we find that $n$ can only take odd numbers.
\end{proof}
So its Lax equations can be defined as
\begin{equation*}
\ \ \ \  \ \ \ \ \ \partial_{2n+1,\alpha}L=[B_{2n+1,\alpha},L],\\ \ \ \ \ \partial_{2n+1,\alpha}R_{\beta}=[B_{2n+1,\alpha},R_{\beta}],\ \ 2n+1\neq 0\ mod\ N,\ \
\end{equation*}
\begin{equation}
\ \ \ \  \ \ \ \ \ \  \ \ \ \sum\limits_{\alpha=1}^{n}\partial_{(2j^{'}+1)N,\alpha}L=0,\ \ \ \ \ \ \ \
\sum\limits_{\alpha=1}^{n}\partial_{(2j^{'}+1)N,\alpha}R_{\beta}=0,\ \ j^{'}=0,1,2,\cdots,\ \ \ \ \
\end{equation}
where $\partial_{2n+1,\alpha}=\frac{\partial}{\partial t_{2n+1,\alpha}}$ and $B_{2n+1,\alpha}=(L^{\frac{2n+1}{N}}R_{\alpha})_{+}$.\\
Taking a dressing operator $\phi=\phi(A\partial^{N})=\sum\limits_{i=1}^{\infty}\alpha_{i}(A\partial^{N})^{-i}\ (\alpha_{0}=I,\ \phi^{*}=\phi^{-1})$, then the wave function of the mcCGD hierarchy is $W(t,z)=\phi e^{\xi(t,z)}=\omega(t,z)e^{\xi(t,z)}$, where
\begin{equation*}
\xi(t,z)=\sum\limits_{i=1}^{\infty}\sum\limits_{\alpha=1}^{n}t_{2i-1,\alpha}E_{\alpha}z^{2i-1}.\ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
In addition, the adjoint wave function of the mcCGD hierarchy is
\begin{equation*}
W^{*}(t,z)=(\phi^{*})^{-1}e^{-\xi(t,z)}=\phi e^{-\xi(t,z)}.\ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
Then we can get the Sato equations $\partial_{2n+1,\alpha}\phi=-(B_{2n+1,\alpha})_{-}\phi\ (2n+1\neq 0\ mod\ N)$ and $\sum\limits_{\alpha=1}^{n}\partial_{(2j^{'}+1)N,\alpha}\phi=0\ (j^{'}=0,1,2,\cdots)$ of the mcCGD hierarchy.\\
And the Lax equations of the mcCGD hierarchy can also be obtained by the compatibility conditions of the following linear partial differential equations
\begin{equation}
\begin{split}
L^{\frac{2n+1}{N}}W=z^{2n+1}W,\ \ \ \ \ \partial_{2n+1,\alpha}W=B_{2n+1,\alpha}W,\ 2n+1\neq 0\ mod\ N,\\
\sum\limits_{\alpha=1}^{n}\partial_{(2j^{'}+1)N,\alpha}W=L^{(2j^{'}+1)}W,\ \ j^{'}=0,1,2,\cdots, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{split}
\end{equation}
where $W(t,z)$ is the wave function of the mcCGD hierarchy.

\subsection{Additional symmetry of the mcCGD hierarchies}
\ \ \ \ From the above knowledge points related to the mcCGD hierarchy, we can find that many of its definitions and properties are similar to those of the mcBGD hierarchy, and the biggest difference lies in the related definitions and operations of their adjoint. Next, we mainly discuss the knowledge of adjoint in detail. Similarly, we first give the Orlov-Shulman's operator $M=\phi\Gamma\phi^{-1}$ of the mcCGD hierarchy, where
\begin{equation*}
\Gamma=\sum\limits_{i=1}^{\infty}\sum\limits_{\alpha=1}^{n}(2i-1)t_{2i-1,\alpha}A^{\frac{2i-2}{N}}\partial^{2i-2}E_{\alpha},\ \ 2i-1\neq 0\ mod\ N,\ \ \ \ \ \ \
\end{equation*}
then we can get
\begin{equation*}
M^{*}=(\phi\Gamma\phi^{-1})^{*}=\phi\Gamma\phi^{-1}=M. \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
After some necessary knowledge reserve, we will give the definition of the additional symmetry of the mcCGD hierarchy. The $C_{l,m,\alpha}$ in the additional symmetry of the mcCGD hierarchy is obviously different from the $D_{l,m,\alpha}$ in the definition of the additional symmetry of the mcBGD hierarchy. The fundamental reason is that their Lax operators have different constraints.
\begin{proposition}
In the C type multi-component Gelfand-Dickey hierarchy, the additional symmetric generators $C_{l,m,\alpha}$ should satisfy $(C_{l,m,\alpha})^{*}=-C_{l,m,\alpha}$, then $C_{l,m,\alpha}$ can be taken as
\begin{equation}
C_{l,m,\alpha}=M^{m}L^{\frac{l}{N}}R_{\alpha}-(-1)^{l}R_{\alpha}L^{\frac{l}{N}}M^{m}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
\end{proposition}
\begin{proof}
According to the the additional symmetry of the mcCGD hierarchy, we can know that
\begin{equation*}
\partial_{l,m,\alpha}^{*}\phi^{*}=(\partial_{l,m,\alpha}^{*}\phi)^{*}=(-(C_{l,m,\alpha})_{-}\phi)^{*}=-\phi^{*}(C_{l,m,\alpha})_{-}^{*},\ \ \
\end{equation*}
and according to $\phi^{*}=\phi^{-1}$, we can deduce that
\begin{equation*}
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \partial_{l,m,\alpha}^{*}\phi^{*}=\partial_{l,m,\alpha}^{*}\phi^{-1}=-\phi^{-1}(\partial_{l,m,\alpha}^{*}\phi)\phi^{-1}=\phi^{-1}(C_{l,m,\alpha})_{-}=\phi^{*}(C_{l,m,\alpha})_{-},
\end{equation*}
by comparing the results of the above two different methods, we can find that
\begin{equation*}
(C_{l,m,\alpha})_{-}=-(C_{l,m,\alpha})_{-}^{*},
\end{equation*}
so $\ C_{l,m,\alpha}=-(C_{l,m,\alpha})^{*}.$\\
Combined with the additional symmetric generators$\ M^{m}L^{\frac{l}{N}}R_{\alpha}\ $of the mcGD hierarchy, we will have
\begin{equation*}
(M^{m}L^{\frac{l}{N}}R_{\alpha})^{*}=R_{\alpha}^{*}(L^{\frac{l}{N}})^{*}(M^{m})^{*}=(-1)^{l}R_{\alpha}L^{\frac{l}{N}}M^{m}, \ \ \ \ \ \ \
\end{equation*}
then it's easy to construct
\begin{equation*}
C_{l,m,\alpha}=M^{m}L^{\frac{l}{N}}R_{\alpha}-(-1)^{l}R_{\alpha}L^{\frac{l}{N}}M^{m}. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
\end{proof}
\begin{definition}
The solution of the differential equation
\begin{equation}
\frac{\partial\phi}{\partial_{l,m,\alpha}^{*}}=-(C_{l,m,\alpha})_{-}\phi,
\end{equation}
where
\begin{equation}
\partial_{l,m,\alpha}^{*}=\frac{\partial}{\partial t_{l,m,\alpha}^{*}},\ \ C_{l,m,\alpha}=M^{m}L^{\frac{l}{N}}R_{\alpha}-(-1)^{l}R_{\alpha}L^{\frac{l}{N}}M^{m},\ \ \ \ \ \ \
\end{equation}
is called the additional symmetry of the C type multi-component Gelfand-Dickey hierarchy.
\end{definition}
Similarly, for the differential operator $C_{l,m,\alpha}$, assuming its negative part disappears and let the operator $(C_{l,m,\alpha})_{+}$ act on $W$, we can get an equation related to $z$
\begin{equation}
(C_{l,m,\alpha})_{+}W=(1-(-1)^{l})z^{l}E_{\alpha}\partial^{m}_{z}W-(-1)^{l}mlz^{l-1}E_{\alpha}\partial^{m-1}_{z}W.
\end{equation}
Note:This system can also be rewritten into a linear equation for the isomonodromy problem.\\
The following is a brief description of the construction method of $C_{l,m,\alpha}$.
According to the additional symmetry of the mcCGD hierarchy, we can deduce
\begin{equation}
\ \ \ \ \ \partial_{l,m,\alpha}^{*}L=-[(C_{l,m,\alpha})_{-},L], \ \ \ \ \ \ \
\partial_{l,m,\alpha}^{*}R_{\beta}=-[(C_{l,m,\alpha})_{-},R_{\beta}].
\end{equation}
For the additional flows of the mcCGD hierarchy, they have a fraction of the flows that can survive.
\begin{theorem}
In the additional flows of the C type multi-component Gelfand-Dickey hierarchy, the flows which satisfy the condition $(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0$ and are shaped like $\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*}$ or the condition $l=2i\ (i\in\mathbb{Z})$ and are shaped like $\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*}$ can survive.
\end{theorem}
\begin{proof}
For the Lax operator $L$ of the mcCGD hierarchy, its negative part is also equal to zero.\\
And we know $\partial_{l,m,\alpha}^{*}L=-[(C_{l,m,\alpha})_{-},L]$, so let's consider
\begin{equation*}
\begin{split}
\ \ \ \ \ \ \ \ \ \ \partial_{l,m,\alpha}^{*}L_{-}=&-[(C_{l,m,\alpha})_{-},L]_{-}=-(\phi[\Gamma^{m}A^{\frac{l}{N}}\partial^{l}E_{\alpha},A\partial^{N}]\phi^{-1})_{-}\\
&+(-1)^{l}(\phi[E_{\alpha}A^{\frac{l}{N}}\partial^{l}\Gamma^{m},A\partial^{N}]\phi^{-1})_{-},
\end{split}
\end{equation*}
after some deductions, the upper formula can be reduced to
\begin{equation*}
\partial_{l,m,\alpha}^{*}L_{-}=\partial_{l,m,\alpha}^{*}L_{-}=(1-(-1)^{l})mN(M^{m-1}L^{\frac{N+l-1}{N}}R_{\alpha})_{-},\ \
\end{equation*}
only when $(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0$ or $l=2i\ (i\in\mathbb{Z})$, we can obtain
\begin{equation*}
\sum\limits_{\alpha=1}^{n}\partial_{l,1,\alpha}^{*}L_{-}=(1-(-1)^{l})mN(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0.\ \ \ \ \ \ \ \
\end{equation*}
\end{proof}
The surviving additional flows of the mcCGD hierarchy satisfy the following propositions.
\begin{proposition}
The additional symmetric flows $\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*}$ which satisfy the condition $(M^{m-1}L^{\frac{N+l-1}{N}})_{-}=0$ or the condition $l=2i\ (i\in\mathbb{Z})$ commute with the flows $\partial_{2k+1,\beta}\ (2k+1\neq 0\ mod\ N)$ of the C type multi-component Gelfand-Dickey hierarchy.
\end{proposition}
\begin{proof}
\begin{equation*}
\ \ \ \ \ \ \ \ \big[\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*},\partial_{2k+1,\beta}\big]\phi=\Big(\sum\limits_{\alpha=1}^{n}\big[\partial_{l,m,\alpha}^{*},\partial_{2k+1,\beta}\big]\Big)\phi=
\sum\limits_{\alpha=1}^{n}\Big(\big[\partial_{l,m,\alpha}^{*},\partial_{2k+1,\beta}\big]\phi\Big),
\end{equation*}
\begin{equation*}
\begin{split}
&[\partial_{l,m,\alpha}^{*},\partial_{2k+1,\beta}]\phi\\
=&-\partial_{l,m,\alpha}^{*}((B_{2k+1,\beta})_{-}\phi)+\partial_{2k+1,\beta}((C_{l,m,\alpha})_{-}\phi)\\
=&[(C_{l,m,\alpha})_{-},(B_{2k+1,\beta})_{-}+\partial_{2k+1,\beta}]_{-}\phi+(B_{2k+1,\beta})_{-}(C_{l,m,\alpha})_{-}\phi\\ \ \ \ \ \ \ \
&+[\partial_{2k+1,\beta},(C_{l,m,\alpha})_{-}]_{-}\phi-(C_{l,m,\alpha})_{-}(B_{2k+1,\beta})_{-}\phi\\
=&[(C_{l,m,\alpha})_{-},(B_{2k+1,\beta})_{-}]\phi-[(C_{l,m,\alpha})_{-},(B_{2k+1,\beta})_{-}]_{-}\phi\\
=&0,
\end{split}
\end{equation*}
then $\big[\sum\limits_{\alpha=1}^{n}\partial_{l,m,\alpha}^{*},\partial_{2k+1,\beta}\big]\phi=0,$ so the proposition is proved.
\end{proof}
After some calculations, we find that whether the odd mcCGD hierarchy or the even mcCGD hierarchy, their string equations are the same. The derivation process is given below. First we consider a additional flow
\begin{equation*}
\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}L^{\frac{2k}{N}}=-\sum\limits_{\alpha=1}^{n}\big[(C_{-(2k-1),1,\alpha})_{-},L^{\frac{2k}{N}}\big],\ 2k=0\ mod\ N,\ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
after a series of calculations, we will obtain
\begin{equation*}
\sum\limits_{\alpha=1}^{n}\partial_{-(2k-1),1,\alpha}^{*}L^{\frac{2k}{N}}=\sum\limits_{\alpha=1}^{n}\big[(C_{-(2k-1),1,\alpha})_{+},L^{\frac{2k}{N}}\big]+4kI,\ \ \ \ \ \
\end{equation*}
When $2k=0\ mod\ N$, $L^{\frac{2k}{N}}$ is a differential operator, then
\begin{equation*}
\sum\limits_{\alpha=1}^{n}\big[L^{\frac{2k}{N}},\frac{1}{4k}(C_{-(2k-1),1,\alpha})_{+}\big]=I,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{equation*}
thus $\Big[L^{\frac{2k}{N}},\frac{1}{4k}\big(\sum\limits_{\alpha=1}^{n}(C_{-(2k-1),1,\alpha})_{+}\big)\Big]=I\ $ is the string equation of the mcCGD hierarchy.
In some literatures, the string equation is described as the form of $[P,Q]=1$, where $P$ and $Q$ are differential operators, which is equivalent to the string equation derived by us, and refers to the condition that the operator is independent of variables.
\\[5pt]

{\bf {Acknowledgements:}}
Chuanzhong Li  is  supported by the National Natural Science Foundation of China under Grant No. 11571192 and K. C. Wong Magna Fund in
Ningbo University.\\


TITLE: If $f$ is one-to-one and continuous on the closed interval $[a,b]$ then prove that $f$ is strictly monotone on $[a,b]$
QUESTION [3 upvotes]: If $f$ is one-to-one and continuous on the closed interval $[a,b]$ then prove that $f$ is strictly monotone on $[a,b]$.

So my plan was to prove this by contradiction.
I'm wondering if there is a simple way to prove this that I am missing. Any help is appreciated!

REPLY [4 votes]: Suppose that $f(a)<f(b)$ we show that $f$ is strictly increasing.
Let $x\in [a,b]$ firstly, we show that $f(x)>f(a)$ if $f(x)\leq f(a)$, since $f$ is continuous, the image of $[x,b]$ by $f$ is an interval, since it contains $f(x)\leq f(a)$ and $f(b)>f(a)$ it contains every element $u$ such that $f(x)\leq u\leq f(b)$ in particular, there exists an element $z\in [x,b]$ such that $f(z)=f(a)$, since $a<x$ we have a contradiction since $f$ is bijective.
Suppose $f$ is not increasing, there exists $x,y\in [a,b] x<y, f(x)\geq f(y)$ if $f(x)=f(y)$ $f$ is not injective, done. Suppose $f(x)>f(y)$, since $f$ is continue, $f([a,x])$ is an interval, it contains every element $u$ such that $f(a)\leq u\leq f(x)$, in particular since $f(a)\leq f(y) <f(x)$ we deduce  that $[a,x]$  contains $z$ such that $f(z)=f(y)$, $z\leq x<y$ this is a contradiction since $f$ is bijective. done.
If $f(a)> f(b)$, the previous part shows that $-f$ is strictly increasing.


TITLE: Collision between a photon and an electron
QUESTION [1 upvotes]: Looking through this AP Physics question, I was struck by how the 'collision' between a photon and electron looks so much like a macroscopic collision. Is this even physically possible?
Look at the last page of this pdf: http://lodischool.tripod.com/dovesol/DOVE02SOL.pdf
EDIT: Some more questions:
How could a photon collide with an electron, when their positions cannot be determined exactly? Also considering how very small the electron is, I doubt that it is even possible to make the two collide; and if it is, how could you possible detect that? It also seems as though the photon and electron are acting as particles, which seems to me not to be the whole story.
What if I put the electrons behind a double slit apparatus, and treat individual photons as particles? Based on this "compton scattering," it's possible for the photon to be deflected any which way. I could claim that the diffraction pattern observed in the double-slit experiment is due to compton scattering, among other factors. Prove me wrong!

REPLY [5 votes]: Here are real events relating to the last page of the pdf link you gave:

Fig.1 This bubble chamber picture shows some electromagnetic events such as pair creation or materialization of high energy photon into an electron-positron pair (green tracks), the Compton effect (red tracks), the emission of electromagnetic radiation by accelerating charges (violet tracks) (bremsstrahlung) and the knock-on electrons or delta ray (blue tracks)

Photons are invisible in bubble chambers as they interact only with direct collisions with electrons, called Compton scattering, or pair production in the field of a nucleus. Charge particles turn in the perpendicular to the plane magnetic field; this allow us to measure their momentum and charge and the ionisation of the tracks allows the identification of masses.
At the lower left of the picture, there is an electron( identified by its ionisation)  which loses energy into a photon, and the photon pair produces some centimeters away, an electron positron pair.
in the middle  right side, we see a positron that loses energy into a photon and the photon kicks an electron from the atoms of the chamber, this is a Compton scatter.   This corresponds to the diagram in the last page of your link, except it has been reduced to one dimension. In reality there are two dimensions, because the photon gives part of its momentum/energy kicking at an angle . The following is the correct diagram kinematically:

It should not be surprising that classical scatters and particle scatters kinematically are the same, because momentum and energy conservation hold both classically and quantum mechanically. It is the probability of interaction that is different in the microcosm of elementary particles to the billiard ball particle scattering. In simple scattering experiments the kinematics are not different ( except that special relativity holds in the microcosm).
Edit after edits in question.
I think the image , from a real experiment, answers whether a photon can hit an electron or not.
Now you ask:

What if I put the electrons behind a double slit apparatus, and treat individual photons as particles? Based on this "compton scattering," it's possible for the photon to be deflected any which way. I could claim that the diffraction pattern observed in the double-slit experiment is due to compton scattering, among other factors. Prove me wrong!

The difraction pattern of individual photons, even when sent one at a time, is a direct result of the quantum mechanical nature of the photon. The solution of the boundary conditions imposed by the two slits gives a probability distribution that displays an interference pattern. Even though electromagnetic interactions viewed as Feynman diagrams are similar, it is the boundary conditions that determine the probability of scatter, and two slits is different than two particle scatter, the fields are different and the solutions are different.

REPLY [2 votes]: They do not need to necessarily collide like balls. I guess the picture in your book is illustrative. Conservation laws apply to any kind of interaction between them. Note that details of the collision are not even provided in the question but you still can calculate the answer.
The detailed theory of photon-electron interactions is called Quantum Electrodynamic.


TITLE: Length of a pivoted rectangle
QUESTION [3 upvotes]: I have a rectangle with a swivel point along the left edge at the midpoint between the top and bottom of the rectangle. The swivel point will be the point "a", the bottom right corner is point "b", the top right corner is point "d" and the point on the right edge equidistant between point "b" and "d" is point "f". I need to know the length from point "a" to point "f" under the follow constraints as the rectangle is pivoted upwards with point "a" being fixed. The original length of the rectangle is unknown but there are some known endpoints.
As the rectangle pivots upwards a triangle will be formed with points "a" "b" and "c". Point "c" is below the rectangle on the line that line segment "af" was originally on prior to pivoting up. The vertex "acb" is a 90-degree angle and the length between points "a" and "c" is known. Along line segment "ac" is point "e". Point "e" forms a triangle with points "a" and "d".The vertex "aed" is a 90-degree angle and the length of "ed" is known. How do I find the length between points "a" and "f"?
I've tried to look at all the triangles formed from the lines. I know if I figured out a single triangle everything else would fall into place. With the parallel lines that the points fall on the problem feels like there should be some known relationships but I can't figure it out.

REPLY [2 votes]: Let $h$ be the intersection between $af$ and $de$, and set: $A=ac$, $B=de$, $C=df$, $X=af$, $\theta=\angle fae$. We have then:
$$
ae=ac-ec=A-2C\sin\theta;\quad
eh=ed-dh=B-{C\over\cos\theta};\quad
ah=af-fh=X-C\tan\theta.
$$
But $eh=ah\sin\theta$ and $ae=ah\cos\theta$, hence we get two equations for the unknowns $X$ and $\theta$:
$$
B=X\sin\theta+C\cos\theta;\quad
A=C\sin\theta+X\cos\theta.
$$
The problem is then reduced to solve the above coupled equations for $X$, but the task doesn't seem so trivial: I tried with Mathematica obtaining a very large expression which cannot be simplified.
EDIT.
Eliminating $\theta$ from the above equations one gets a single quartic equation in X:
$$
X^4-(A^2 + B^2 + 2C^2)X^2 + 4ABCX + C^2(C^2- A^2 - B^2) =0.
$$
But a general formula for the solution is very large. Better to substitute there the values of $A$, $B$, $C$ and solve numerically.


TITLE: Cannot find the limit of this integral
QUESTION [2 upvotes]: I am trying to find the limit of the following integral as $n$ grows to infinity.
$$\int_1^{ n+1 }\frac { \ln x }{ x^n } \, dx $$
However, when I am doing an integration by parts, the result looks ugly. It is not in a simple form from which, I think, one can find the limit in a simple way (indeterminate form, I mean).
So there should be a trick in there to find the limit.
Can someone help me on this one please?
Regards,

REPLY [0 votes]: \begin{align}
\int_{1}^{n + 1}{\ln\left(x\right) \over x^{n}}\,{\rm d}x
&=
\lim_{m \to 0}
{\partial \over \partial m}\int_{1}^{n + 1}{x^{m} \over x^{n}}\,{\rm d}x
=
\lim_{m \to 0}
{\partial \over \partial m}
\left\lbrack\left(n + 1\right)^{m - n + 1} - 1 \over m - n + 1\right\rbrack
\\[3mm]&=
\lim_{m \to 0}
{\left(n + 1\right)^{m - n + 1}\ln\left(n + 1\right)\,\left(m - n + 1\right)
 -
 \left\lbrack\left(n + 1\right)^{m - n + 1} - 1\right\rbrack
 \over
\left(m - n + 1\right)^{2}}
\\[3mm]&=
{\left(n + 1\right)^{1 - n}\,\ln\left(n + 1\right)\,\left(1 - n\right)
 -
 \left(n + 1\right)^{1 - n} + 1
 \over
\left(n - 1\right)^{2}}
\\[5mm]--------&-------------------------------
\end{align}
\begin{align}
\lim_{n \to \infty}\int_{1}^{n + 1}{\ln\left(x\right) \over x^{n}}\,{\rm d}x
&=
\lim_{n \to \infty}
{n^{-n}\,\ln\left(n\right)\,\left(-n\right) - n^{-n} + 1 \over n^{2}}
\\[3mm]&=
-\lim_{n \to \infty}{\ln\left(n\right) \over n^{n + 1}}
+
\lim_{n \to \infty}{1 \over n^{n + 2}} + \lim_{n \to \infty}{1 \over n^{2}}
\\[5mm]&
\end{align}
$$
\begin{array}{|c|}\hline
\\
\color{#ff0000}{\large\quad%
\lim_{n \to \infty}\int_{1}^{n + 1}{\ln\left(x\right) \over x^{n}}\,{\rm d}x
=
0\quad}
\\ \\
\hline
\end{array}
$$


text:
\begin{document}

\author[C. P.~Curry]{Clinton P. Curry}

\title[Irreducible Julia Sets]{Irreducible Julia Sets of Rational
  Functions}

\address[Clinton P.~Curry]
{Department of Mathematics\\
  University of Alabama at Birmingham\\
  Birmingham, AL 35294-1170} \email{clintonc@uab.edu}
\urladdr{http://www.math.uab.edu/~curry} \date{\today}

\thanks{The author was supported in part by NSF-DMS-0353825.}

\subjclass[2000]{Primary 37F10, Secondary 54F15}

\begin{abstract}
  We prove that a polynomial Julia set which is a finitely irreducible
  continuum is either an arc or an indecomposable continuum.  For the
  more general case of rational functions, we give a topological model
  for the dynamics when the Julia set is an irreducible continuum and
  all indecomposable subcontinua have empty interior.
\end{abstract}
\maketitle
\thispagestyle{empty}

\section{Introduction}
\label{sec:introduction}

Let $\rsphere$ denote the Riemann sphere, and let $R:\rsphere
\rightarrow \rsphere$ be a rational function.  The \emph{Fatou set}
$F(R)$ is the domain of normality of the iterates $\set{R^i \, \mid \, i
\in \nats}$.  The \emph{Julia set} $J(R)$ is $\rsphere \setminus
F(R)$, which is generally regarded as the subset of $\rsphere$ where
$R$ is chaotic.  If the degree of $R$ is at least two, then $J(R)$ is
a non-empty, compact, perfect subset of $\rsphere$.  A
\emph{continuum} is a non-empty, compact, and connected metric space.
A connected Julia set of a rational function is therefore a
subcontinuum of $\rsphere$.

This paper is motivated by recent work that addressed a conjecture of
P. M.~Ma\-ki\-en\-ko.  The exact statement is not important to the
subject of this paper; interested parties are referred to
\cite{Curry:ul} for more information.  What is important is the
conclusion: We proved that, if $R$ is a rational function that is a
counterexample to Makienko's conjecture, its Julia set $J(R)$ is a
\emph{finitely irreducible continuum}, and hence an
\emph{indecomposable continuum}.

\begin{defn}[Irreducible]
  A continuum $X$ is \emph{irreducible about a set $A \subset X$} if
  no proper subcontinuum of $X$ contains $A$.  If $X$ is irreducible
  about a finite subset, it is called \emph{finitely irreducible}.  If
  $X$ is irreducible about a two-point subset, it is simply called
  \emph{irreducible}.
\end{defn}
\begin{rem}\label{rem:lc_irred}
  The unit interval $[0,1]$ is an irreducible continuum, since no
  proper subcontinuum contains both of its endpoints. For the same
  reason, a finite tree is a finitely irreducible continuum.
  Conversely, any locally connected finitely irreducible continuum is a
  finite tree, since locally connected continua are also arcwise
  connected.  The arc is the only locally connected irreducible
  continuum.
\end{rem}
\begin{figure}
  \centering
  \includegraphics[height=4cm]{sin1x}
  \hspace{1cm}
  \includegraphics[height=4cm]{knaster_continuum}
  \caption{On the left is the $\sin \frac1x$ continuum, the simplest
    non-locally connected irreducible continuum.  On the right is the
    Knaster continuum, the simplest indecomposable continuum.}
  \label{fig:sin1x}
\end{figure}

It is well-known that there are Julia sets of rational functions which
are irreducible continua.  For example, the Julia set of the
polynomial $z \mapsto z^2-2$ is the interval $[-2,2]$.  Though it is
not known if every Julia set which is an irreducible continuum is an
arc, there are far more complicated examples of irreducible continua.

\begin{defn}[Indecomposable]
  A continuum $X$ is \emph{decomposable} if there are proper
  subcontinua $A$ and $B$ such that $A \cup B = X$.  Otherwise, $X$ is
  \emph{indecomposable}.
\end{defn}

\begin{rem}\label{rem:indec}
  Indecomposable continua are also irreducible continua, even strongly
  so.  If $X$ is an indecomposable continuum, then for a residual set
  of pairs $(x,y) \in X \times X$, $X$ is irreducible between $x$ and
  $y$.
\end{rem}

The arc and indecomposable continua represent the extremes of
topological complexity for irreducible continua.  The goal of this
paper is to prove that these two cases are representative for rational
Julia sets which are finitely irreducible continua.  We prove in
\prettyref{sec:polynomial-case} that if $P$ is a polynomial such that
$J(P)$ is an irreducible continuum, then $J(P)$ is either an arc or an
indecomposable continuum.  We conjecture the same result holds for
rational functions, but we only prove a weaker structure theorem for
rational functions using the same tools (see
\prettyref{sec:invariance}).

The argument naturally divides into two cases.  The first case is when
the Julia set in question is an irreducible continuum which has an
indecomposable subcontinuum with non-empty interior relative to the
Julia set.  Not much is said about the Julia set in this case, either
topologically or dynamically.  The other case is that the Julia set
contains no indecomposable subcontinuum with interior, which is the
case that most of the statements in this paper address.  To avoid
awkward repetition, we introduce the following terminology.

\begin{term}
  A continuum which is finitely irreducible such that no
  indecomposable subcontinuum has interior will be called a
  \emph{finished continuum}.
\end{term}

\begin{ack}
  I would like to express my appreciation for what Professor Devaney
  has done for the field and for the reserachers who work in it.  I
  would also like to thank my doctoral advisors, Dr. Alexander Blokh
  and Dr. John C.~Mayer, for helpful conversations on this and other
  topics.
\end{ack}

\section{Aposyndesis and Vought's Decomposition}
\label{sec:aposyndesis}

In this section, we review the notion of \emph{aposyndesis} due to
Jones \cite{Jones:1941db, Jones:1952nx} and an associated
decomposition defined by Vought \cite{Vought:1974qy}.  The goal is to
define a monotone map $m:J(R) \rightarrow Y$, where $Y$ is a
topologically simpler space.  In our case, it will happen that $Y$ is
locally connected, and studying $Y$ will help us gain insight about
$J(R)$.

\begin{defn}[Aposyndesis]
  A continuum $X$ is aposyndetic at a point $p \in X$ if every point
  $q \in X \setminus \set{p}$ is contained in the interior (relative to
  $X$) of a continuum $Y \subset X \setminus \set{p}$.
\end{defn}
Aposyndesis is a topological property which is slightly weaker than
local connectivity.  It is not difficult to see that any locally
connected continuum is aposyndetic.  Conversely, any aposyndetic
continuum which is the boundary of a simply connected domain in
$\rsphere$ is locally connected \cite[Theorem 14]{Whyburn:1939zr}.
Since the Julia set of a polynomial is the boundary of the domain of
attraction of $\infty$, an aposyndetic polynomial Julia set is also
locally connected.  A simple example of a non-locally connected
aposyndetic continuum is illustrated in \prettyref{fig:aposyndetic}.

Associated to the concept of aposyndesis is the set-valued function
$T$, defined below.  It should be considered in analogy with a
topological closure operator, except that it is not necessarily
idempotent.

\begin{figure}
  \centering
  \includegraphics[width=5cm]{cone}
  \caption{The suspension over a convergent sequence.  This continuum
    is aposyndetic, but not locally connected.}
  \label{fig:aposyndetic}
\end{figure}
\begin{defn}[The set-valued function $T$]
  For a non-empty set $A \subset X$, let $T(A)$ be the set of $x \in
  X$ for which all subcontinua of $X \setminus A$ containing $x$ have
  empty interior relative to $X$.  A set $A$ is called
  $\emph{$T$-closed}$ if $T(A) = A$.
\end{defn}
\begin{rem}\label{rem:T_properties}
  It follows immediately from the definition that, for any sets $A
  \subset B$, $A \subset T(A) \subset T(B)$.  If $X$ is the $\sin
  \frac 1 x$ continuum depicted in \prettyref{fig:sin1x} and $A$ is a subset
  of the limit bar, that $T(A)$ equals the entire limit bar.  Further,
  because every proper subcontinuum of an indecomposable continuum is
  nowhere dense, $T(A) = X$ whenever $X$ is an indecomposable
  continuum and $A$ is a subset.
\end{rem}

Vought showed that the set-valued function $T$ can be used to define a
continuous monotone map from any finished continuum $X$ onto a finite
tree $Y$.  He does so giving a partition $\mathcal D(X)$ of $X$ into
subcontinua. The space $Y$ is defined as the quotient space $X /
\mathcal D(X)$.  In general, $m$ is monotone when each element of
$\mathcal D(X)$ is connected, and $m$ is continuous when the
collection $\mathcal D(X)$ is upper semi-continuous.  The reader is
referred to \cite[Chapter 3]{Nadler:1992ek} for these and other facts
about decompositions of continua and continuous maps.

The following theorem describes the map defined by Vought.
\begin{thm}[Theorem 1 of \cite{Vought:1974qy}]\label{thm:decomp}
  Let $M$ be a finished continuum which is irreducible about $n$
  points, but no fewer.  Let
  \[\mathcal D(M) = \set{ T^n\{x\} : x \in M }\]
  where $T^n$ denotes the $n$-fold composition of $T$. Then the
  following hold:
  \begin{enumerate}
  \item $\mathcal D(M)$ is an upper semi-continuous decomposition of $M$,
  \item the elements of $\mathcal D(M)$ are continua,
  \item the quotient space $M / \mathcal D(M)$ is locally connected, and
  \item each element of $\mathcal D(M)$ has no interior in $M$.
  \end{enumerate}
  Further, $\mathcal D(M)$ is the only partition satisfying these
  properties.  Also, the quotient $M / \mathcal{D}(M)$ is a tree with
  $n$ endpoints.
\end{thm}

\begin{rem}\label{rem:trans_to_maps}
  This can be translated as the existence of a continuous monotone map
  $m:M \rightarrow Y$ where $Y$ is a finite tree with $n$ endpoints
  and point inverses are nowhere dense in $M$.
\end{rem}

For the remainder of the paper, when $X$ is a finished continuum the
symbol $\mathcal D(X)$ will denote the decomposition provided by this
theorem.  For a point $x \in X$, let $D(x)$ be the element of
$\mathcal D(X)$ containing $x$.  The following observation relates the
properties of $T$-closed sets to the decomposition $\mathcal D(X)$.

\begin{lem}\label{lem:closed_saturated}
  Let $X$ be a finished continuum irreducible about $n$ points.  Then
  every $T$-closed set is a union of elements of $\mathcal D(X)$.
\end{lem}
\begin{proof}
  Suppose that $A \subset X$ is $T$-closed and let $x \in A$.  By
  \prettyref{rem:T_properties}, $x \in T^n(\set{x}) \subset T^n(A)$,
  which equals $A$ since $A$ is $T$-closed.  But $T^n(\set{x}) \in \mathcal
  D(X)$ and $x \in A$ was arbitrary, so $A = \bigcup_{x \in A}D(x)$.
\end{proof}

\section{The Decomposition and the Dynamics}
\label{sec:invariance}
The goal of this section is to prove the following theorem.
\begin{thm}\label{thm:main_rat}
  Suppose that $R$ is a rational function whose Julia set is a
  finitely irreducible continuum.
  \begin{enumerate}
  \item $J(R)$ contains an indecomposable subcontinuum with non-empty
    interior relative to $J(R)$, or
  \item $J(R)$ is irreducible between two points, and admits a
    monotone map $m:J(R) \rightarrow[0,1]$ such that
    \begin{enumerate}
    \item $m^{-1}(x)$ is nowhere dense for all $x \in [0,1]$, and
    \item $m$ semiconjugates $R|_{J(R)}$ to an open, topologically
      exact, at most $\operatorname{deg}(R)$-to-one map
      $g:[0,1]\rightarrow[0,1]$.
    \end{enumerate}
  \end{enumerate}
\end{thm}

The main work is to show that the decomposition $\mathcal{D}(J(R))$,
defined in the previous section, respects the dynamics of $R$ when
$J(R)$ is a finished continuum.  Specifically, we aim to show that the
image under $R$ of any element of $\mathcal D(J(R))$ is again an
element of $\mathcal D(J(R))$.

First we prove two lemmas relating $T$ and $R$.
\begin{lem}\label{lem:preimage_apo}
  Let $R$ be a rational map.  If $A$ is $T$-closed in $J(R)$, then
  $R^{-1}(A)$ is as well.
\end{lem}
\begin{proof}
  Let $A \subset Y$ be a $T$-closed set, and let $A^{-1}=R^{-1}(A)$.
  Suppose $x \notin A^{-1}$; we will show that $x \notin T(A^{-1})$.

  Note that $R(x) \notin A$ since $x \notin T(A^{-1})\supset A^{-1}$.
  Since $A$ is $T$-closed, we have that $R(x) \notin T(A)$.  Let $H
  \subset Y \setminus A$ be a continuum containing $R(x)$ in its
  interior.  Then $R^{-1}(H)$ is a closed set that is also a
  neighborhood of $x$.  Let $H^{-1}$ be the component of $R^{-1}(H)$
  containing $x$.  Note that $R^{-1}(H)$ is the union of finitely many
  continua, so $H^{-1}$ is open in $R^{-1}(H)$, therefore containing a
  neighborhood (in $J(R)$) of $x$.  Also,

  \begin{align*}
    H^{-1} \cap A^{-1} &\subset R^{-1}(H) \cap R^{-1}(A) \\
    &= R^{-1}(H \cap A) \\
    &= \emptyset.
  \end{align*}
  Thus $x \notin T(A^{-1})$.  Since $x$ was any point in $X \setminus
  A^{-1}$, we see that $T(A^{-1})=A^{-1}$.
\end{proof}

\begin{thm}\label{thm:elements_to_elements}
  Let $R$ be a rational function.  If the Julia set $J$ is a finished
  continuum, then for each $D \in \mathcal{D}(J(R))$ we have $R(D) \in
  \mathcal{D}(J(R))$.
\end{thm}

\begin{proof}
  Let $Y = J(R) / \mathcal D(J(R))$, and let $m:J(R) \rightarrow Y$ be
  the associated quotient map.  Recall from \prettyref{thm:decomp}
  that elements of $\mathcal D(J(R))$ are nowhere dense in $J(R)$ and
  that $Y$ is a finite tree.  For $x \in J(R)$, recall that $D(x)$
  denotes the unique element of $\mathcal D(J(R))$ containing $x$.  We
  must show that $R(D(x)) = D(R(x))$.  Let $K$ be the component of
  $R^{-1}(D(R(x)))$ containing $x$.  Because $R$ is confluent, we see
  that $R(K) = D(R(x))$.  It is then sufficient to show that $D(x) =
  K$.

  We see that $K$ has empty interior in $J(R)$, since $R(K) = D(R(x))$
  has empty interior in $J(R)$ and $R|_{J(R)}$ is an open map.  As a
  component of a $T$-closed set, $K$ is $T$-closed \cite[Lemma
  2.6]{FitzGerald:1967rz} and the union of elements of $\mathcal
  D(J(R))$ by \prettyref{lem:closed_saturated}.  Therefore, $K =
  m^{-1}(m(K))$, so $m(K)$ must have empty interior in $Y$.  However,
  $Y$ is a finite tree, so the only subcontinua with empty interior
  are points.  We then conclude that $m(K)$ is a point, and therefore
  $K$ is an element of $\mathcal{D}(J(R))$.
\end{proof}

Now we study the induced map $g:J(R)/\mathcal{D}(J(R)) \rightarrow
J(R)/\mathcal{D}(J(R))$ and use it to draw conclusions about the set
$J(R)$ itself.

\begin{thm}\label{cor:induced_map}
  Suppose that $R$ is a rational map whose Julia set is a finished
  continuum.  Then the monotone map $m:J(R) \rightarrow
  J(R)/\mathcal{D}(J(R))$ is a monotone semiconjugacy of $R|_{J(R)}$
  to a map $g:J(R)/\mathcal{D}(J(R)) \rightarrow
  J(R)/\mathcal{D}(J(R))$ of a finite tree.  The map $g$ is then open,
  topologically exact, and at most $\operatorname{deg}(R)$-to-one.
\end{thm}
\begin{proof}
  For brevity, set $Y = J(R) / \mathcal{D}(J(R))$.  Notice that $m
  \circ R$ is constant on every set of the form $m^{-1}(y)$ by
  \prettyref{thm:elements_to_elements}.  Therefore, the map $g:Y
  \rightarrow Y$ defined by $g = m \circ R \circ m^{-1}$ is a
  single-valued, continuous map (see \cite[Theorem
  22.2]{Munkres:1975ad}).

  To see that $g$ is open and topologically exact, let $U \subset Y$
  be open.  Then $m^{-1}(U)\subset J(R)$ is an open set, saturated
  with respect to the quotient map $m$.  Since $R|_{J(R)}$ is open,
  $R(m^{-1}(U))\subset J(R)$ is open, and by
  \prettyref{thm:elements_to_elements} this set is also saturated.
  Therefore, $m(R(m^{-1}(U)))=g(U) \subset Y$ is open, so $g$ is an
  open map.  Also, since $m^{-1}(U) \subset J(R)$ is open, there
  exists $n \in \nats$ such that $R^n(m^{-1}(U))=J(R)$.  Thus $m \circ
  R^n \circ m^{-1}(U)=g^n(U) = Y$, so $g$ is topologically exact.

  To show that $g$ is at most $\operatorname{deg}(R)$-to-one, let $y
  \in Y$. Then $R^{-1}(m^{-1}(y))$ is the union of at most
  $\operatorname{deg}(R)$ elements of $\mathcal{D}(J(R))$, since each
  element must map onto $m^{-1}(y)$ and hence must contain a preimage
  of $y$.  Therefore, $m(R^{-1}(m^{-1}(y)))=g^{-1}(y)$ has cardinality
  at most $\operatorname{deg}(R)$.
\end{proof}

\begin{cor}\label{cor:arc}
  If $R$ is a rational function and $J(R)$ is a finished continuum,
  then $J(R) / \mathcal D(J(R))$ is an arc, and $J(R)$ is irreducible.
\end{cor}

\begin{proof}
  Suppose that $J(R) / \mathcal{D}(J(R))$ has a cut point $x$ of order
  at least $3$.  Since $g$ is topologically exact, we have that $B =
  \bigcup_{n \in \nats}g^{-n}(x)$ is dense in
  $J(R)/\mathcal{D}(J(R))$.  Since $g$ is open, each point of $B$ is
  also a branch point of $J(R) / \mathcal{D}(J(R))$.  Therefore, $J(R)
  / \mathcal{D}(J(R))$ has infinitely branch points if it has one.  A
  finite tree does not have infinitely many branch points, so $J(R) /
  \mathcal{D}(J(R))$ is an arc.  That $J(R)$ is irreducible follows
  from \prettyref{thm:decomp} and that the arc is an irreducible
  continuum.
\end{proof}

\begin{rem}\label{rem:nsaw}
  It is known that topologically exact open maps of intervals are
  conjugate to $n$-saw maps, which are of the form
  \[
  f(x)=\begin{cases}
    (n-2i)x & \mbox{if $x \in [2i, 2i+1]$}\\
    (2(i+1)-n)x & \mbox{if $x \in [2i+1, 2i+2]$}.
  \end{cases}
  \]
  Therefore, $g$ is conjugate to a map of this sort.
\end{rem}

Combining the above, we can prove the main theorem of the section.

\begin{proof}[Proof of \prettyref{thm:main_rat}]
  Suppose that $J$ contains no indecomposable subcontinuum with
  interior.  The monotone map $m$ is the quotient map corresponding to
  the decomposition of \prettyref{thm:decomp}.  That $m$ is an arc is
  \prettyref{cor:arc}, and the facts about the induced map $g:I
  \rightarrow I$ are from \prettyref{cor:induced_map}.
\end{proof}

\section{The Polynomial Case}
\label{sec:polynomial-case}

In the case of polynomial Julia sets, we can say more.
\begin{thm}
  Let $P$ be a polynomial, and suppose that $J(P)$ is finitely
  irreducible.  Then either
  \begin{enumerate}
  \item $J$ is an indecomposable continuum, or
  \item $J$ is homeomorphic to an arc.
  \end{enumerate}
\end{thm}

\begin{proof}
  Suppose that $J$ is not indecomposable.  Then no indecomposable
  subcontinuum has interior in $J$ \cite[Theorem 1]{Childers:2006fk},
  so $J(P)$ is a finished continuum.  Let $m:J \rightarrow [0,1]$ be
  the monotone map provided by \prettyref{thm:main_rat}, and let $g:I
  \rightarrow I$ be the map to which $P|_{J(P)}$ is conjugate via $m$.
  We will show that $J(P)$ is homeomorphic to an arc by showing it is
  locally connected (see \prettyref{rem:lc_irred}).  This will be
  accomplished by showing that the forward orbits of its critical
  points are finite \cite[Theorem 19.7]{Milnor:2006fr}.

  For $t \in [0,1]$, let $J_t$ denote the union of the continuum
  $m^{-1}(t)$ with its bounded complementary domains.  It is not
  difficult to see that $P(J_t)=J_{g(t)}$, and that the map
  $P|_{J(t)}:J_t \rightarrow J_{g(t)}$ is open for each $t \in I$ (for
  instance by \cite[Lemma 13.13]{Nadler:1992ek}).

  Extend $m:J(P) \rightarrow [0,1]$ to a map $\hat m:\bigcup_{t \in
    [0,1]} J_t \rightarrow [0,1]$ by sending points of $J_t$ to the
  point $t \in [0,1]$.  Each $\hat m^{-1}(t)$ is a non-separating
  plane continuum, so the Vietoris-Begle theorem implies that $\hat m$
  induces an isomorphism between the \v{C}ech cohomologies of
  $\bigcup_{t \in [0,1]} J_t$ and $[0,1]$.  Therefore, $\bigcup_{t \in
    [0,1]}J_t$ is a non-separating plane continuum containing $J(P)$,
  so $\bigcup_{t \in [0,1]} J_t$ equals the filled Julia set.  Since
  $J(P)$ is connected, each critical point is contained in the filled
  Julia set \cite[9.5]{Milnor:2006fr} and therefore in some $J_t$.

  We now show that a point $t \in [0,1]$ is a critical point of $g$ if
  and only if $J_t$ contains a critical point of $P$.  Suppose that a
  fiber $J_t$ does not contain any critical point of $P$.  Then there
  is a (saturated) neighborhood $U$ of $J_t$ on which $P$ is a
  homeomorphism.  In particular, no two fibers contained in $U$ have
  the same image, so $g$ is a homeomorphism on $m(U)$.

  On the other hand, suppose that $t$ is not a critical point of $g$.
  Then there is a (saturated) neighborhood $U$ of $J_t$ such that no
  two fibers contained in $U$ have the same image.  If any fiber
  $J_{t_0}$ does not map by a homeomorphism onto its image, then
  $J_{t_0}$ must contain a critical point since it and its image are
  full continua.  Let $V \subset U$ be a disk about $J_t$ such that
  $R|_{V \setminus J_t}$ is a covering map onto its image.  Then a
  fiber which does not contain a critical point intersects $V$, and
  $R$ is one-to-one in a neighborhood of it.  Therefore, $R|_{V}$ is a
  homeomorphism, and $J_t$ cannot contain a critical point.

  If one takes a saturated open set $V \subset U$ about $J_t$ such
  that $V \setminus J_{t_0}$ does not contain the preimage of any
  critical value, we see that $R|_V$ is a homeomorphism, so $J_t$
  contains no critical point of $R$.

  Notice that $J_0 \cup J_1$ is a forward-invariant set.  Suppose
  without loss of generality that $P(J_0) = J_0$.  Note that $J_0$
  cannot contain a critical point of $P$, since $0$ cannot be a
  critical point of $g$.  Therefore, $P|_{J_0}$ is a forward expanding
  homeomorphism, which implies that $J_0$ must be a point \cite[Lemma
  18.8]{Milnor:2006fr}.  Then $J_1$ must also be a point, either
  because it is fixed or because it maps to $J_0$.  Because every
  critical point of $g$ must map into $\set{0,1}$, every critical
  point of $R$ must map into $J_0 \cup J_1$ and hence has a finite
  orbit.  This implies that $J(P)$ is locally connected, and therefore
  an arc.
\end{proof}

\section{Further Work}
The main novelty in this work is the use of what might be called a
\emph{locally connected model} of the Julia set to draw conclusions
about its topology and the dynamics of its rational map.  There is
more structure on polynomial Julia sets than there are on rational
Julia sets, so it is to be expected that more can be concluded in the
polynomial case.  Locally connected models for polynomial Julia sets
have been studied further in joint work with Alexander Blokh and Lex
Oversteegen in \cite{Blokh:2008il}, where we characterized the
\emph{finest locally connected model} for the action of a polynomial
on its Julia set.

Vought's decomposition is rather special, and only applies to finitely
irreducible continua.  It is an example of a broader notion called a
\emph{core decomposition}.  A decomposition is core with respect to a
property P if it has property P and it refines all decompositions with
property P.  Therefore, Vought's decomposition for finished continua
is core with respect to the property that the quotient space is
locally connected.

Generally speaking, there is no core decomposition of an arbitrary
continuum with locally connected quotient.  However, there is always a
core decomposition with an aposyndetic
quotient \cite{FitzGerald:1967rz}, of which Vought's decomposition is
a special case. Such a model may serve in the stead of a locally
connected model where locally connected models are unavailable.
However, the utility is limited when the continuum in question
contains an indecomposable subcontinuum with interior, as it must be
absorbed into an element of any decomposition to a nice space.  There
are several pertinent questions.

\begin{ques}
  Does there exist a rational function whose Julia set contains an
  indecomposable continuum with interior?
\end{ques}
\begin{ques}
  Does there exist a rational function whose Julia set does not have a
  finest locally connected model?
\end{ques}
\begin{ques}
  Let $R$ be a rational function with connected Julia set.  Is the
  finest decomposition to an aposyndetic continuum invariant with
  respect to $R$?
\end{ques}
\begin{ques}
  For what useful topological properties $P$ does there exist a finest
  decomposition of every Julia set $J(R)$ satisfying $P$?  Is the
  decomposition dynamic?  Which of these is the appropriate
  analog for the finest locally connected model?
\end{ques}

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