text:
\begin{document}
\title{Stochastic properties of an inverted pendulum on a wheel on a soft surface.}
\author{O.M.~Kiselev}
\maketitle
\begin{abstract}
We study dynamics of the inverted pendulum on the wheel on a soft surface and under a proportional-integral-derivative controller. The behaviour of such pendulum is modelled by a system with a differential inclusion. If the the system has a sensor for the rotational velocity of the pendulum, the tilt sensor and the encoder for the wheel then this system is observable. The using of the observed data for the controller brings stochastic perturbations into the system.  The  properties of the differential inclusion under stochastic control is studied  for upper position of the pendulum.  The formula for the time, which  the pendulum spends near the upper position, is derived.
\end{abstract}

\section{Introduction}
\label{secIntroduction}

The wheeled inverted pendulum  (WIP) is a popular model for studies nor only  dynamics and the system  of the control for robotics equipments near instability positions. A list of contemporary works in this field is too large. Here we should mention studies for the derivation of the mathematical model for the WIP and the control synthesis, which one can see for example in the book \cite{Formalskii2016Eng}, and the article  \cite{MartynenkoFormalskii2013Eng}. The questions concerned an stability and control for WIP with two wheels were considered in  \cite{PathakFranchAgrawal2005} for the horizontal and in  \cite{SamiMichalskaAngeles2007} for the inclined surface.

\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[ultra thick]

\draw (-1,0)--(-0.35,0.1);
\draw (0.55,0.2)--(1.5,0.4);
\draw[thin] (1.5,0.4)--(2.5,0.6);
\draw[thin] [->] (2.4,0) arc[start angle=0, end angle=13, radius=2.4];
\draw (2.6,0.3) node {$z$} ;

\foreach \i in {-1,-0.7,-0.4,-0.1,0.2,0.5,0.8,1.1}
	\draw (\i,\i*0.13+0.15)--(\i+0.3,\i*0.13+0.15-0.3);

\draw[fill=white] (0,1) circle [radius=1];

\draw(0,1)--(1,4);
\draw(0,1) circle [radius=0.1];

\draw[fill=white] (1,4) circle [radius=0.5];

\begin{scope}[thin]
\draw (0,0) -- (2.5,0);
\draw (2.1,-0.3) node {$x$} ;
\draw (0,0) --(0,5);
\draw (-0.2,5.1) node {$y$} ;
\draw[<-] (0,3) arc[start angle=90,end angle=77,radius=3];
\draw (0.3,3.3) node {$\alpha$} ;
\draw [->](1.3,1) arc[start angle=0,end angle=60,radius=1.3];
\draw (1.3,2) node{$\beta$};
\end{scope}
\end{tikzpicture}
\end{center}
\caption{The inverted pendulum on the wheel. Let us denote  $r$ is the radius of the wheel, $l$ is the pendulum length, $\alpha$ is the angle of the pendulum turn, $\beta$ is the angle of the wheel turn and $z$ is the inclination angle of the surface.}
\label{figurePendulumOnTheWheel}
\end{figure}

We apply the proportional-integral-derivative (PID) controller to stabilize the WIP at  the upper point.  We should mention the PID controller is often used to the objects of different nature \cite{AstromHagglund1994}. For example the PID controller can be used to stabilize the WIP on the hard horizontal, see \cite{HatakeyamaShimada2008},\cite{NasirRajaIsmailAhmad2010} and a review   \cite{ChanStolHalkyard2013}.  But the soft surface is more complicated to stabilize the WIP. The mathematical model for the WIP controlled by  the PID controller on the soft surface  was offered and detailed studied in \cite{Kiselev2020} (see also preprint \cite{Kiselev2020arxiv1}).

The major part of the PID controller is an observation value of the angle of the pendulum position. This angle can be found by gyroscope sensor . Usually the sensor obtains the value with a small stochastic error. The maximal amplitude and the dispersion for the error is  standardized by specification of the sensor, see for example  \cite{mpu6000}.

An additional sensor for the WIP is a tilt sensor. Such sensors are commonly used and have a detailed specification in which the interval of errors and the dispersion are pointed, see for example  \cite{MMA7361L}.

To obtain amount of the rotation of the wheel we use an encoder. If the wheel does not slip, the errors of the digital encoder appear due to round up only. Slipping brings additional errors to the value of the encoder. Therefore one of the problems for the controller is to detect the slipping.

In this work we show that the gyroscope and tilt for the pendulum and encoder for the wheel are enough to observe the state of the system in framework of the mathematical model.

However the noise of the sensors and the slipping bring stochastic errors into the value of the sensors. Therefore the controller which uses these values has stochastic perturbations. Hence the mathematical model with the digital PID controller is stochastic.

In section \ref{secEquationsOfMotion} we describe the mathematical model of WIP on the soft surface with the PID controller. The dynamic model on the soft surface contains a rolling resistance. Pure mathematically this resistance is described by  the differential inclusion,
see \cite{Kiselev2020}.

In section \ref{secObservation} one can find the formulas for the current state of the WIP. These formulas use the data obtained from the gyroscope,  tilt sensor and encoder.

In section \ref{secStochasticPropOfObservedData} we discuss the stochastic properties of the data for the PID controller. The errors and dispersion of the data from the sensors are assumed as known form the technical notes.

In section \ref{secStochasticPropOfPendulum} we consider the properties of the WIP under the PID controller with stochastic perturbations. The perturbations appear in the control by using the data with the stochastic errors.

\section{Dynamical system for the WIP}
\label{secEquationsOfMotion}

Here we will consider the moving of the WIP with the additional control torque on the wheel. The torque is denoted by $u$. Let us assume that the equipment moves on soft surface with inclination $z$. The value of $z$ depends on the traversed path  by the wheel and hence one can write  $z=z(\beta)$. The mathematical model of such moving has the form (see \cite{Kiselev2020}, also preprint \cite{Kiselev2020arxiv1}):
\begin{eqnarray}
\ddot{\alpha}
&=&
 \sin(\alpha) -(\cos(\alpha-z)\ddot{\beta}+\sin(\alpha-z)\dot{\beta}^2)\rho
-2\frac{\rho}{\zeta}u,
\nonumber\\
(\zeta +2)\rho\,\ddot{\beta}
& \in &
F(\alpha,\dot{\alpha},\ddot{\alpha},\dot{\beta}).
\label{inclusionForPendulumOnWheelFullWithStabAndDump}
\end{eqnarray}
Let us denote
$$
f=-\sin(z)-\left(\ddot{\alpha}\cos(\alpha-z) -\dot{\alpha}^2\sin(\alpha-z)\right)  \zeta+\frac{2}{\rho}u.
$$
In formula  (\ref{inclusionForPendulumOnWheelFullWithStabAndDump}) the map  $F(\alpha,\dot{\alpha},\ddot{\alpha},\dot{\beta})$ has the form:
$$
F(\alpha,\dot{\alpha},\ddot{\alpha},\dot{\beta})
=
\left\{
\begin{array}{cc}
f-\nu\ \sgn(\dot{\beta}),& \quad\quad \{\forall\dot{\beta}\not=0\};
\\
(-\nu,\nu), & \quad\{\dot{\beta}=0\}\cup\{ |f|\le\nu\};
\\
f, & \quad \{\dot{\beta}=0\}\cup\{\{\alpha,\dot{\alpha},\ddot{\alpha}\} \in \{|f|>\nu\}\}.
\end{array}
\right.
$$
Here the parameters of the mathematical model are following: $\alpha$  is an angle of the pendulum turn, $\beta$ is the angle of wheel turn, $z$ is the current inclination of the soft surface, $\nu$ is the torque of the friction resistant, $\rho=r/l$ is the ratio of the wheel radius and the length of the pendulum, $\zeta$ is the ratio of pendulum mass and the rim mass.

The system (\ref{inclusionForPendulumOnWheelFullWithStabAndDump}) can be simplify for hard  ($\nu=0$) surface with the constant inclination ($z\equiv\epsilon$). As a result one gets the second order equation for $\alpha$:
\begin{eqnarray}
(\sin^2(\epsilon-\alpha)\zeta +2)\ddot{\alpha}=
(2+\zeta)\sin(\alpha)  +
\sin(\epsilon)  \cos( \epsilon -\alpha) +
\nonumber\\
\frac{1}{2}\zeta\dot{\alpha}^2 \sin(2(\epsilon -\alpha))-
2\left(
\frac{1}{\rho}\cos(\epsilon-\alpha)+
\left(1+\frac{2}{\zeta}\right)\rho
\right) u.
\label{EqPendulumOnWheel1WithStab}
\end{eqnarray}
The particular case for the WIP on the hard horizontal ($\epsilon=0$) looks as follow:\begin{eqnarray}
(\sin^2(\alpha)\zeta+2)\ddot{\alpha}
&=
&
(2+\zeta)\sin(\alpha)-\frac{1}{2}\zeta \dot{\alpha}^2\sin(2\alpha) -
\nonumber
\\
&\,&
2\left(\frac{1}{\rho}\cos(\alpha)+\left(1+\frac{2}{\zeta}\right)\rho\right)u
\label{EqPendulumOnWheel0WithStab}
\end{eqnarray}

The control torque with the PID controller has the following form:
$$
u=k_1\alpha+k_2\dot{\alpha}+k_3 A,\quad \text{where},\quad A\equiv\int^t \alpha(t) dt.
$$

In this case system  (\ref{inclusionForPendulumOnWheelFullWithStabAndDump})  has a particular solution:
\begin{eqnarray}
\alpha
\equiv0,
&
\quad
\displaystyle
A=\frac{\sgn(\dot{\beta})\zeta\nu\rho}
{\left(2k_3\zeta+4k_3\right)\rho^2+2k_3\zeta},
\nonumber\\
\beta
=
\bigg\{
&
\begin{array}{c}\displaystyle
\beta_0+\beta_1(t-t_0)-\frac{\zeta\nu\rho\,\sgn(\dot{\beta})}
{\left(2k_3\zeta+4k_3\right)\rho^2+2k_3\zeta}
\frac{(t-t_0)^2}{2},\quad (t-t_0)<T;
\\
\displaystyle
\beta_0+\beta_1 T-
\frac{\zeta\nu\rho\,\sgn(\dot{\beta})}
{\left(2k_3\zeta+4k_3\right)\rho^2+2k_3\zeta}
\frac{T^2}{2},\quad (t-t_0)\ge T,
\end{array}
\label{partialSolution}
\end{eqnarray}
where
$$
T=\,\frac{1}{\beta_1}\,\frac{\zeta\nu\rho\,\sgn(\dot{\beta})}
{\left(2k_3\zeta+4k_3\right)\rho^2+2k_3\zeta},\quad
\{t_0,\beta_0,\beta_1\}\in\mathbb{R}.
$$
There exists the set of the parameters  $\zeta,\rho,k_1,k_2,k_3$ when solution  (\ref{partialSolution}) is an attractor as  $(t-t_0)<T$ \cite{Kiselev2020} (see also preprint \cite{Kiselev2020arxiv1}).

In an ideal case the control should be defined by the current  values of $\alpha,\dot{\alpha},A$, but for real equipment these parameters can be obtained using the sensors at  the moment $t_i$, where $i\in\mathbb{N}$. As a result the control is a discrete function: $u(t)=u(t_i)=u_i$.

At the interval $t\in(t_i,t_{i+1})$  the control torque  $u_i$  is a constant. Such system has a first integral and can be integrate in quadratures.

For example the moving on the hard surface with the constant inclination (\ref{EqPendulumOnWheel1WithStab}) has a first integral at the interval  $t\in(t_i,t_{i+1})$:

\begin{eqnarray}
\mathcal{E}_i=
&
\cos(\alpha)(\zeta +2)
+\sin(\epsilon)\sin(\epsilon-\alpha)+\left(\frac{1}{2}
\,\sin^2(\epsilon-\alpha)\zeta+1\right)\dot{\alpha}^2+
\nonumber\\
&
\left(2\left(1+\frac{2}{\zeta}\right)\rho\alpha -\frac{2}{\rho}\sin(\epsilon-\alpha))\right)u_i
.
\label{formulaForConservationLawOnSlantedSurface}
\end{eqnarray}

This formula allows us to integrate  $\dot{\alpha}$ at the interval $t\in(t_i,t_{i+1})$ and we can  write the parameters of the system at  $t=t_{i+1}$:
\begin{eqnarray*}
\alpha_{i+1}=F_1(\alpha_i,\dot{\alpha}_i,u_i,dt),
\\
\dot{\alpha}_{i+1}=F_2(\alpha_i,\dot{\alpha}_i,u_i,dt),
\\
A_{i+1}=F_3(\alpha_i,\dot{\alpha}_i,u_i,dt)
\end{eqnarray*}

One can obtain the first integral for WIP on the hard horizontal surface (\ref{EqPendulumOnWheel0WithStab}) if one assumes  $\epsilon=0$.

The mathematical model for the WIP on the soft surface does not integrate obviously. Nevertheless this model can be written in the form:
\begin{eqnarray*}
\dot{A}=\alpha,\quad
\dot{\alpha}=a,\\
\dot{a}=\sin(\alpha)-(\cos(\alpha-z)\dot{b}+\sin(\alpha-z)b^2)\rho-2\frac{\rho}{\zeta}u,\\
\dot{\beta}=b,\quad
\dot{b}\in\frac{1}{(2+\zeta)\rho}F(\alpha,a,\dot{a},b).
\end{eqnarray*}

One can obtain the numeric solution of this differential inclusion at the interval $t\in(t_i,t_{i+1})$. Let us define the map:
$$
(A_n,\alpha_n,a_n,\beta_n,b_n)
\to
(A_{n+1},\alpha_{n+1},a_{n+1},\beta_{n+1},b_{n+1}).
$$ Formally this map can be written like a discrete dynamical system:
\begin{eqnarray*}
\mathbf{X}_{n+1}=\mathbf{F}(\mathbf{X}_n),\quad \text{where}\quad  \mathbf{X}_n=(A_n,\alpha_n,a_n,\beta_n,b_n,u_n).
\end{eqnarray*}

\section{Observability of the mathematical model for WIP}
\label{secObservation}

In this section we consider the set of the data necessary for the observability of the parameters of the mathematical model for WIP  (\ref{inclusionForPendulumOnWheelFullWithStabAndDump}).

The angle of the tilt for the pendulum is defined by gyroscope. The gyroscope can be work in two different cases. The first one it defines the angle of the pendulum and the second one it define the angle velocity for the pendulum. In the second case one should integrates the angle velocity to obtain the pendulum angle. Below we will use the gyroscope in the mode of angle velocity. This means the value of the angle velocity  $\dot{\alpha}$ is known at the moment of the measurement.

Besides the gyroscope we assume as existing the tilt sensor. This sensor define the linear acceleration of the pendulum in the plane of the moving of WIP.

Let us define the coordinates as  $(x,y)$, where $x$ is the horizontal coordinate and $y$ is the vertical one. The projections of the acceleration vector on the coordinate axes $Ox$ and $Oy$ one can write as follows:
\begin{eqnarray*}
\ddot{x}&=&\ddot{\beta}r\cos(z(\beta))+
\ddot{\alpha}l\sin(\alpha),
\\
\ddot{y}&=&\ddot{\beta}r\sin(z(\beta))+
\ddot{\alpha}l\cos(\alpha)+g.
\end{eqnarray*}
It is convenient to write these formulas in the form:
\begin{eqnarray*}
\ddot{\alpha}l\cos(\alpha+z)&=&-\ddot{x}\sin(z)+
\ddot{y}\cos(z)-g\sin(z),
\\
\ddot{\beta}r\cos(\alpha+z)&=&\ddot{x}\cos(\alpha)-
\ddot{y}\sin(\alpha)+g\sin(\alpha).
\end{eqnarray*}

One more sensor is the encoder. This sensor allows us to define the turn of the wheel. The data from the encoder allow to obtain the mean value of the angle velocity of the wheel as value of the difference between the current value of the turn angle of the wheel and another one value at previous measurement:
$$
\dot{\beta}\sim \frac{\beta(t)- \beta(t-\Delta t)}{\Delta t}.
$$

Let us consider the system for WIP on the horizontal surface (i.e. $z=0$):
$$
\ddot{\alpha}l\cos(\alpha)=\ddot{y}
$$
$$
\ddot{\beta}r\cos(\alpha)=\ddot{x}\cos(\alpha)-
\ddot{y}\sin(\alpha)+g\sin(\alpha).
$$
The value $a_1=\dot{\alpha}$ is known from the sensor. Let us define by $a_2(t)=\ddot{y}/l$, $b_2=\ddot{x}/r$, $\gamma=g/l$  and $b_1=\dot{\beta}$. Then the dynamical system  (\ref{inclusionForPendulumOnWheelFullWithStabAndDump})  can be written as the system of the trigonometric equation and the inclusion:
\begin{eqnarray}
\frac{\mathit{a_2}}{\cos{\left( \alpha \right) }}=-\left( \cos{\left( \alpha -z\right) } \left( \frac{\sin{\left( \alpha \right) } \gamma }{\cos{\left( \alpha \right) } \rho }-\frac{\mathit{a_2} \sin{\left( \alpha \right) }}{\cos{\left( \alpha \right) } \rho }+\mathit{b_2}\right) +{\mathit{b_1^2}} \sin{\left( \alpha -z\right) }\right)  \rho
\nonumber
\\
-\frac{2 u \rho }{\zeta }+\sin{\left( \alpha \right) },
\label{EqForObservationAlphaU}
\\
\left( \zeta +2\right) \, \left( \frac{\sin{\left( \alpha \right) } \gamma }{\cos{\left( \alpha \right) } \rho }-\frac{\mathit{a_2} \sin{\left( \alpha \right) }}{\cos{\left( \alpha \right) } \rho }+\mathit{b_2}\right)  \rho
\in
\left\{
\begin{array}{cc}
f-\nu\sgn(b_1),\quad b_1\not=0;
\\
(-\nu,\nu),\quad b_1=0 \cup |f|<\nu;
\\
f,\quad b_1=0 \cup |f|\ge\nu;
\end{array}
\right.
\label{InclusionForObservationAlphaU}
\end{eqnarray}
where
$$
f=
\frac{2 u}{\rho } -
\left( \frac{\mathit{a_2} \cos{\left( \alpha -z\right) }}{\cos{\left( \alpha \right) }}-{\mathit{a_1^2}} \sin{\left( \alpha -z\right) }\right)  \zeta -\sin{(z)}.
$$
The angle of the pendulum $\alpha$ and the control torque
$u$ are the unknown variables in the system  (\ref{EqForObservationAlphaU}), (\ref{InclusionForObservationAlphaU}).

If $b_1\not=0$ or $b_1=0 \cup |f|\ge\nu$ then the inclusion (\ref{InclusionForObservationAlphaU}) turn to the following equation:
\begin{eqnarray*}
\left( \zeta +2\right) \, \left( \frac{\sin{\left( \alpha \right) } \gamma }{\cos{\left( \alpha \right) } \rho }-\frac{\mathit{a_2} \sin{\left( \alpha \right) }}{\cos{\left( \alpha \right) } \rho }+\mathit{b_2}\right)  \rho
=
\\
\frac{2 u}{\rho } -
\left( \frac{\mathit{a_2} \cos{\left( \alpha -z\right) }}{\cos{\left( \alpha \right) }}-{\mathit{a_1^2}} \sin{\left( \alpha -z\right) }\right)  \zeta -\sin{(z)}-\nu\sgn(b_1).
\end{eqnarray*}

As a result one get the system of the equations for  $\alpha,u$. The control torque $u$ can be easy found through the trigonometric functions of  $\alpha$ and hence one get the trigonometric equation for $\alpha$ .

As $\{b_1=0\} \cup \{|f|<\nu\}$ the angle of the pendulum should be solution of the inequality:
$$
\frac{-\nu}{\zeta+2}-b_2\rho< (\gamma -\mathit{a_2})\tan(\alpha) <\frac{\nu}{\zeta+2}-b_2\rho.
$$
Here one get the observed parameter  $u$. To obtain the integral term $A$ of the PID controller one should use the following formula:
$$
A=u-\frac{k_1}{k_3}\alpha-\frac{k_2}{k_3}a_1.
$$

\begin{theorem}
Let one know the values of the acceleration   $(\ddot{x},\ddot{y})$, angle velocity of the pendulum  $\dot{\alpha}$ and angle velocity of the wheel $\dot{\beta}$, then the observed dynamical system is solution of the trigonometric equation  (\ref{EqForObservationAlphaU}) and the inclusion (\ref{InclusionForObservationAlphaU}).
\end{theorem}
The equations for small values of  $\nu,\alpha,\ddot{y},\dot{\alpha},\ddot{x},\dot{\beta}$ and ($z\equiv0$) can be written in the following form:
\begin{eqnarray}
\mathit{a_2}
\sim
-\frac{2 \rho  u}{\zeta }+\left(1-\gamma\right)  \alpha -\rho \, \mathit{b_2},
\nonumber\\
(\zeta +2)(  \rho \, b_2+  \gamma  \alpha)
\in
\left\{
\begin{array}{cc}
\sim\frac{2 u}{\rho } -\zeta a_2-\sgn(b_1)\nu, \quad b_1\not=0;
\\
\sim(-\nu,\nu),\quad  b_1=0 \cup |-\zeta a_2+2u/\rho|<\nu;
\\
\sim\frac{2 u}{\rho } -\zeta a_2,\quad b_1=0\cup|-\zeta a_2+2u/\rho|\ge\nu.
\end{array}
\right.
\label{InclusionForSmallParameters}
\end{eqnarray}

\begin{corollary}
The important case for the WIP on the hard horizontal is more simplest.  In particular the angle $\alpha$ is the solution of the equation:
\begin{eqnarray}
\rho^2\sin{\left( \alpha \right) } \left( 2 \gamma  \zeta -2 \mathit{a_2} \zeta +4 \gamma -4 \mathit{a_2}\right) +
\nonumber\\
\zeta\sin{\left( 2 \alpha \right) } \left( -{\mathit{a_1^2}} {{\rho }^{2}}+{{\mathit{b_1}}^{2}} \rho +\gamma -\mathit{a_2}-1 \right) =
\\
\rho^2\cos{\left( \alpha \right) } \left( -2 \mathit{b_2} \zeta \, \rho -4 \mathit{b_2}\,\rho-2 \mathit{a_2} \zeta \right) -
\nonumber\\
\mathit{b_2} \cos{\left( 2 \alpha \right) } \zeta  \rho
-\mathit{b_2} \zeta  \rho -2 \mathit{a_2}\, \zeta
\nonumber
\label{EqForObservedAlphaOnSolidHorizontal}
\end{eqnarray}
\end{corollary}

For small values of  $\nu,\alpha,\ddot{y},\dot{\alpha},\ddot{x},\dot{\beta}$ we get:

$$
\alpha\sim -\frac{\left( {{\rho }^{2}}+1\right)  \zeta \, \mathit{a_2}+\left( \left( {{\rho }^{3}}+\rho \right)  \zeta +2 {{\rho }^{3}}\right) \, \mathit{b_2}}{\left( \left( {{\rho }^{2}}+1\right)  \gamma -1\right)  \zeta +2 {{\rho }^{2}} \gamma },
$$
$$
u\sim
-\frac{\left( \rho \, {{\zeta }^{2}}+2 \rho  \gamma  \zeta \right) \, \mathit{a_2}+\left( {{\rho }^{2}}\, {{\zeta }^{2}}+2 {{\rho }^{2}} \zeta \right) \, \mathit{b_2}}{\left( \left( 2 {{\rho }^{2}}+2\right)  \gamma -2\right)  \zeta +4 {{\rho }^{2}} \gamma }.
$$

\section{The observability and the stochastic properties}
\label{secStochasticPropOfObservedData}

\begin{figure}
\includegraphics[scale=0.45]{figCalculated_and_simulated_of_observed_Beta_DBeta.pdf}
\caption{
The observed phase curve on the plane  $(\beta,\dot{\beta})$ are black. This curve is obtained using the data from the sensors as $z\equiv0$. The white line is the result of numeric solution for the system for WIP on soft surface. The   feedback controller use the tilt sensor and the gyroscope (\ref{inclusionForPendulumOnWheelFullWithStabAndDump}). The parameters of the system are following: $\rho=0.2,\zeta=10,\, \nu=0.05,\gamma=1$, the PID coefficients are: $k_1=1.7, k_2=0.2, k_3=0.02$. The relative errors are uniform distributed data at the interval $(-0.02,0.02)$. The dynamic system
(\ref{inclusionForPendulumOnWheelFullWithStabAndDump}) solved at $A\sim 0.2385,\,\alpha=0.02$, $\dot{\alpha}=0,\,\beta=0,\dot{\beta}=0.5$ by Runge-Kutta method of fourth-order method with the step $0.1$.
}
\label{figSolutionAndObservation}
\end{figure}

The value of the controlling torque at $t_i$ is defined by the measurement of observed values of the parameters of the dynamic system.  The obtained data from the sensors  and the computed observed data at  $t_i$ will be denoted by variables with upper symbol $\breve{\bullet}$.

The absolute errors and the dispersion are known for typical sensors. Below we will assume that we know the standard deviation $\sigma_j$ of the measured data at $t=t_j$.

Let the measured data be following:
\begin{eqnarray*}
\breve{a}_1=\dot{\alpha}+\delta^{(1)},\,
\breve{a}_2=\ddot{y}/l+\delta^{(2)},\,
\breve{b}=\beta+\delta^{(3)},\,
\breve{b}_1=\dot{\beta}+\delta^{(4)},\,
\breve{b}_2=\ddot{x}/r+\delta^{(5)}.
\end{eqnarray*}
Here $\delta^{(i)}$ is stochastic error.

Let us consider the WIP on the hard horizontal ($\nu=0$ and $z\equiv0$). We will assume the errors are small and one can use a linear system for find observable values of $\alpha,u$.
\begin{eqnarray*}
\frac{\mathit{a_2}}{\cos{\left( \alpha \right) }}=-\left( \cos{\left( \alpha \right) } \left( \frac{\sin{\left( \alpha \right) } \gamma }{\cos{\left( \alpha \right) } \rho }-\frac{\mathit{a_2} \sin{\left( \alpha \right) }}{\cos{\left( \alpha \right) } \rho }+\mathit{b_2}\right) +{\mathit{b_1^2}} \sin{\left( \alpha \right) }\right)  \rho -
\\
\frac{2 u \rho }{\zeta }+\sin{\left( \alpha \right) }
\\
\left( \zeta +2\right) \, \left( \frac{\sin{\left( \alpha \right) } \gamma }{\cos{\left( \alpha \right) } \rho }-\frac{\mathit{a_2} \sin{\left( \alpha \right) }}{\cos{\left( \alpha \right) } \rho }+\mathit{b_2}\right)  \rho =\frac{2 u}{\rho } -\left( \mathit{a_2}-{\mathit{a_1^2}} \sin{\left( \alpha \right) }\right)  \zeta
\end{eqnarray*}

One can derive the equation for $\alpha$:
\begin{eqnarray}
&
\{((-\gamma +a_1^2\cos(\alpha)+a_2)\zeta -2\gamma+2a_2)\rho^2 -
\nonumber
\\
&
b_1^2\cos(\alpha)\zeta\rho + a_2\cos(\alpha) \zeta \}
\sin(\alpha)=
\nonumber
\\
&
(b_2 \cos(\alpha) \zeta +2 b_2\cos(\alpha))\rho^3+ (\sgn(b_1)\cos(\alpha)\nu +a_2\cos(\alpha)\zeta) \rho^2
+
\nonumber
\\
&
b_2\cos(\alpha)^2 \zeta\rho +a_2\zeta.
\end{eqnarray}

For small $\ddot{y},\alpha,\dot{\alpha},\ddot{x},\dot{\beta},\nu$ we obtain the formulas for errors of the observed data $\alpha$ and $A$:
$$
\breve{\alpha}\sim\alpha-
\frac{((\zeta+2)\rho^2+\zeta)\rho}
{(\zeta+2)\gamma\rho^2+(\gamma-1)\zeta}
\delta^{(5)}+
\frac{(\rho^2 +1)\zeta}{(\zeta+2)\gamma\rho^2+(\gamma-1)\zeta}
\delta^{(2)}
$$

\begin{eqnarray}
\breve{u}\sim u+
\frac{(\gamma -1)\zeta\nu\rho }{\left( 2 \gamma  \zeta +4 \gamma \right) \, {{\rho }^{2}}+\left( 2 \gamma -2\right)  \zeta }
(\sgn(b_1+\delta_4)-\sgn(b_1))
\nonumber
\\
-
\frac{(\zeta^2+2\gamma \zeta )\rho }{\left( 2 \gamma  \zeta +4 \gamma \right) \, {{\rho }^{2}}+\left( 2 \gamma -2\right)  \zeta }\delta^{(2)}
-\frac{(\zeta^2+2\zeta)\rho^2}{\left( 2 \gamma  \zeta +4 \gamma \right) \, {{\rho }^{2}}+\left( 2 \gamma -2\right)  \zeta }\delta^{(5)},
\nonumber
\end{eqnarray}
hence:
$$
\breve{A}\sim \breve{u}-\frac{k_1}{k_3}\breve{\alpha}-
\frac{k_2}{k_3}(\dot{\alpha}+\delta^{(1)}).
$$

{\bf Remark.}While the WIP moves on the soft horizontal the stochastic layer appears near the hyperplane $\dot{\beta}=0$. The width of this layer is  $\min\{\delta^{(4)}\}\le\dot{\beta}\le\max\{\delta^{(4)}\}$. In this layer the stochastic error can be  $\pm\nu$ when $|f|>\nu$. It is important the value of this errors defines by value of the rolling resistance for the wheel and does not depend on the error of the encoder.

The data with stochastic errors are used in the PID controller. As a result the stochastic perturbations appear in the mathematical model for the WIP  (\ref{EqPendulumOnWheel0WithStab} and in the systems  (\ref{EqPendulumOnWheel1WithStab}) and (\ref{inclusionForPendulumOnWheelFullWithStabAndDump}). Therefore the mathematical model with the PID controller looks like the stochastic differential inclusion  (\ref{inclusionForPendulumOnWheelFullWithStabAndDump}).
In partial the results for the observed values of  $(\beta,\dot{\beta})$ and $\alpha$ with stochastic errors are showed in the figures  \ref{figSolutionAndObservation} and  \ref{figAlphaPIDSolidHorizontalFiltered}.

\begin{figure}
\includegraphics[scale=0.5]{figCalculated_and_simulated_of_observed_Alpha.pdf}
\caption{
On the picture one can see the result of numeric modelling for the angle of the pendulum of WIP on the soft horizontal. The horizontal axis defines values of the time and the vertical axis defines the value of the angle of the pendulum. The curve 2 is the numeric value and the curve 1 shows the model of the observed data. The observed data  $\breve{\ddot{\alpha}}_i$ and $\breve{\dot{\alpha}}_i$ are modelled using current values of $\ddot{\alpha}_i$ and $\dot{\alpha}_i$ with the uniform distribution of the relative error at the interval $(-0.05,0.05)$. The value of the angle  $\breve{\alpha}_i$ is defined as the observed calculated using the angle acceleration and angle velocity from the equation  (\ref{EqForObservedAlphaOnSolidHorizontal}). The values  $\breve{\alpha}$ and $\breve{\dot{\alpha}}$ were used to obtain  $\breve{A}$ by integrating by the trapezoidal rule. The  value of the control torque   $u_{i+1}$ was obtained at the interval $t\in(t_i,t_{i+1})$. The system of the equations was solved at the interval $t\in(t_i,t_{i+1})$ with the constant value of the control torque  $u=u_{i+1}$ by Runge-Kutta fourth-order method with the step equals by $0.1$.
}
\label{figAlphaPIDSolidHorizontalFiltered}
\end{figure}

{\bf The remark about filtering data}

The current values of the parameters of the dynamic system for WIP one can obtain by the different approaches.

The first one is the integration of the differential inclusion as the predetermined process. Such approach gives the errors at any step of the integration because of two causes. First of all this errors appear because of the errors in the initial data on the first step of integration. One more cause of the appearance of the errors is the inaccuracy of the mathematical model.

Another one approach is to use the observability of this system. This case does not needed to integrate the differential inclusion. But the errors appears in the current moment because of the errors of the measurement of the data using the sensors.

To minimize the quadratic deviation of the data one can combine the observed data and the forecasted data using the deterministic methematical model. Such algorithms are called as the filters. The filters for the linare system are  well-knowing, see  \cite{Kalman1960}, \cite{KalmanBucy1961}, \cite{BrammerSiffling1975D}. FOr nonlinear smooth systems like the WIP on the hard surface is convenient the generalized Kalman's filter, see \cite{AthansWishnerBertolini1968}, \cite{AustinLeondes1981}. But for the considered here case of WIP on the soft surface the generalized Kalman's filter is  not appropriated because of non-lineared the dinamical system in the neighbouhood of the hypersurface  $\dot{\beta}=0$. One of the opportunity to use  filtering in such case is the sigma-point filter, see \cite{JulierUhlrnannDurrant-Whyte1995}, \cite{JulierUhlmann1997}.

\section{Stochastic properties for WIP on soft horizontal}
\label{secStochasticPropOfPendulum}

\begin{figure}
\hspace{-1.5cm}
\includegraphics[scale=0.4]{figAlpha_under_discrete_control.pdf}
\hspace{-1.5cm}
\includegraphics[scale=0.4]{figBeta_DBeta_under_discrete_control-1.pdf}
\caption{
In this picture one can see the result of the numeric modelling for the behaviour of the angle for the pendulum at $\xi=10$, $\rho=0.2$, $\nu=0.05$, $\gamma=1$, $k_1=1.7$, $k_2=0.2$, $k_3=0.02$. On the left picture the horizontal axis shows the time variable $t$ and the vertical axis shows the angle of the pendulum $\alpha$. On the right picture the horizontal axis shows $\beta$ and the vertical axis shows $\dot{\beta}$. The step of the change of the    control torque is $0.1$. The line is the solution under the discrete control. The results of the measurements  $\breve{\ddot{\alpha}}_i$, $\breve{\dot{\alpha}}_i$ and $\breve{\dot{\beta}}_i$ are modelled by the current values $\ddot{\alpha}_i$, $\dot{\alpha}_i$ and $\dot{\beta}_i$ with the uniform distribution of the  relative errors at the interval $(-0.003,0.003)$. The value of the angle $\breve{\alpha}_i$ is defined as the observed data through the angle acceleration and the angle velocity using (\ref{inclusionForPendulumOnWheelFullWithStabAndDump}). The value $\breve{A}$ is computed using $\breve{\alpha}$, $\breve{\dot{\alpha}}$ integrating by the trapezoidal method. It allows to obtain the control torque $u_{i+1}$ at the interval $t\in(t_i,t_{i+1})$, where $t_{i+1}-t_i=dt$. At $t\in(t_i,t_{i+1})$ the system for the WIP on the soft horizontal is solved for the constant value the control torque $u=u_{i+1}$ by the Runge-Kutta method of the fourth order with the step equals $0.01$.
}
\label{figBetaDBetaPIDStochastic}
\end{figure}

Let the interval $dt$ between the moments of the measurements be small. Then one can see at the dynamic system as a determined dynamic system  (\ref{inclusionForPendulumOnWheelFullWithStabAndDump}) with stochastic perturbation. The stochastic perturbation is contained in the control torque:
$$
\tilde{u}_i= k_1\tilde{\alpha}_i +k_2\tilde{\dot{\alpha}} +k_0\tilde{A}_i.
$$

In the work \cite{Kiselev2020} (see also preprint \cite{Kiselev2020arxiv1}) it was shown that the unperturbed dynamical system with the PID controller has the attractor as $\sgn(\dot{\beta})=\pm1$. This attractor is a line belonged the fifth-dimensional phase space:
$(A,\alpha,\dot{\alpha},\beta,\dot{\beta})=(A_{\pm},0,0,0,\dot{\beta})$, where
$$
\frac{2}{\rho}k_0A_{\pm}=\pm\nu.
$$

On this line the system for WIP is unstable and due to the perturbations crosses to the trajectory with changing of the signum of rotation of the wheel from  $\sgn(\dot{\beta})=\pm1$ to $\sgn(\dot{\beta})=\mp1$. As a result the numeric modelling gives the trajectory like the hysteresis loop, see \cite{Kiselev2020}.

Let us consider here the impact of the stochastic perturbation on the stability for the hysteresis loop. The typical trajectory for the system with the stochastic perturbation is shown on the figure \ref{figBetaDBetaPIDStochastic}.

\begin{theorem}
The line $(A_{\pm},0,0,0,\dot{\beta})$ as $\sgn(\dot{\beta})=\pm1$ is the attractor for the stochastic system (\ref{inclusionForPendulumOnWheelFullWithStabAndDump}).
\end{theorem}

This theorem is corollary from the results of  \cite{Kiselev2020} concerning the stability of the line $(A_{\pm},0,0,0,\dot{\beta})$ as $\sgn(\dot{\beta})=\pm1$ for pure determined dynamical system for WIP under the PID controller and the theorem about stability under constantly perturbations \cite{Krasovskii1959Eng}.

The layer $|\dot{\beta}|\le \max\{\delta^{(3)}\}$ appears in the stochastic system near the hyperplane  $\dot{\beta}=0$. In this layer the term $\nu\sgn(\dot{\beta})$ takes the random values  $\pm\nu$ at  $t\in(t_i,t_{i+1})$.

There exists the small neighbourhood ($\Delta_{\pm}$) near the unstable lines $(A_{\pm}$, $0$, $0$, $0$, $\dot{\beta})$, where  can be obtained four typical cases:

\begin{itemize}
\item Let $\dot{\beta}>0$, $A<A_{+}$
\begin{itemize}
\item and $\sgn{\breve{\dot{\beta}}}=1$, then the trajectory is kept in the neighbourhood of the line $(A_{+},0,0,0,\dot{\beta})$;
\item  and $\sgn{\tilde{\dot{\beta}}}=-1$, then the trajectory is kept in the neighbourhood of the line $(A_{+},0,0,0,\dot{\beta})$.
\end{itemize}
\item Let $\dot{\beta}<0$, $A>A_{-}$
\begin{itemize}
\item and $\sgn{\breve{\dot{\beta}}}=-1$, then the trajectory is kept in the neighbourhood of the line $(A_{-},0,0,0,\dot{\beta})$;
\item  and $\sgn{\breve{\dot{\beta}}}=1$, then the trajectory is kept in the neighbourhood of the line $(A_{-},0,0,0,\dot{\beta})$.
\end{itemize}
\end{itemize}

The sequence of the changes of the trajectories at the neighbourhoods of the lines $(A_{\pm},0,0,0)$ leads to the appearance of the hysteresis loop at the phase plane  $(\beta,\dot{\beta})$, see figure \ref{figSolutionAndObservation}.

Here it is important for applications the average time,which the WIP spends in the neighbourhood of the upper position.

The time between the sequence measurements is equal  $dt$. Let the trajectory be in the neighbourhood  $\Delta_{\pm}$ of the unstable line.  The probability of   $\sgn(\delta^{(3)})=\pm1$ in primary order as $\Delta_\pm\to0$ equals   $p_{\pm}\sim1/2$.  The average time for trajectory in this neighbourhood is following:
$$
T_0=dt\sum_{n=1}^\infty\frac{n}{2^{n}}=2dt.
$$

\begin{theorem}
The average time spending at $\Delta$-neighbourhood of the unstable lines $(A_{\pm},0,0,0,\dot{\beta})$ for the stochastic system  (\ref{inclusionForPendulumOnWheelFullWithStabAndDump}) equals  $2dt$, where $dt$ is the time between the sequenced measurements of the state for the system.
\end{theorem}

\section{Conclusion}
The system for the WIP with discrete control by the PID controller is stochastic due to the errors of the measurements. The stabilising of the WIP on the soft surface leads to the appearance of the hysteresis loop in the plane of the phase variables $\beta,\dot{\beta}$. The average time spending near the upper position was calculated.

config: arxiv


TITLE: Notation question on the colon symbol
QUESTION [3 upvotes]: I'm reading through the first chapter of Ahlfors's Complex Analysis book, and during the section on stereographic projections, he says that we can map any $z = x+iy \in \mathbb{C}$ onto the unit sphere in three dimensions injectively using the equation $z= \frac{x_1+ix_2}{1-x_3}$.
He then says that $x:y:-1= x_1:x_2:x_3-1$, which implies that $(x,y,0)$, $(x_1,x_2,x_3)$, and $(0,0,1)$ all lie on the same line.
My question is what the colon symbol means in this context. I only remember it being used in the context of sets as a replacement for the "|" symbol.

REPLY [2 votes]: The colon notation $\rm\ (x\::\:y\::\:z)\ $ denotes a point in projective space (vs. comma notation $\rm\ (x\:,\:y\:,\:z)\ $ for points in affine space).$\ $ The notation highlights the fact that $\rm\  (x\::\:y\::\:z)\ =\ (\lambda\:x\::\:\lambda\:y\::\:\lambda\:z)\:,\ $ which explains the equality stated by Ahlfors. See for example Kedlaya: Projective Geometry.


TITLE: Prove that if tangent of $x^2+y^2=a^2$ is $px+qy=1$, then $(p, q)$ is on a circle
QUESTION [1 upvotes]: It's actually my textbook problem so I have seen a proof in my class but that doesn't satisfy me. The proof was given by the condition that if $px+qy=1$ is a tangent then the distance from the center $(0,0)$ is equal to radius $a$. From there we find an equation like $p^2 + q^2 = 1/a^2$. As it is an equation of a circle, $(p, q)$ is situated on a circle.
Yes I get the theoretical proof but I don't understand how it happens in real, in a cartesian plane. Please explain me how it would look like in a graph paper. What does the question even mean?

REPLY [3 votes]: Let $(x',y')$ be the point of contact,
then the equation of the tangent $L$ is $x'x+y'y=a^{2}$.
$L$ : $\displaystyle \frac{x'x}{a^{2}}+\frac{y'y}{a^{2}}=1$
$\displaystyle \therefore (p,q)=
 \left(
   \frac{x'}{a^{2}},\frac{y'}{a^{2}}
 \right)$
Since $(x',y')$ lies on the circle, $x'^{2}+y'^{2}=a^{2}$
$\displaystyle \implies p^{2}+q^{2}=\frac{x'^{2}}{a^{4}}+\frac{y'^{2}}{a^{4}}=\frac{a^{2}}{a^{4}}$
i.e. $(p,q)$ lies on the circle $\displaystyle x^{2}+y^{2}=\frac{1}{a^{2}}$


text:
\begin{document}

	\begin{abstract}
Suppose that $\Omega$ is a complex lattice that is closed under complex conjugation and that $I$ is a small real interval, and that $D$ is a disc in $ \mathbb{C}$. Then the restriction $\wp|_D$ is definable in the structure $(\bar{\mathbb{R}},\wp|_I)$ if and only if the lattice $\Omega$ has complex multiplication. This characterises lattices with complex multiplication in terms of definability.

	\end{abstract}

\maketitle
	\section{Introduction}
Since the 1980's model theorists have studied expansions of the real ordered field $\bar{\mathbb{R}}=(\mathbb{R},<,+,\cdot,0,1)$ by various analytic functions. A major result in this area was due to Wilkie, who in \cite{wilkieexp} proved the model completeness of $\rexp=(\bar{\mathbb{R}},\exp)$ where $\exp$ is the real exponential function and this lead to much further activity. \par In particular in \cite{bianconi} Bianconi used Wilkie's ideas together with a functional transcendence result due to Ax to show that no arc of the sine function is definable in $\rexp$. If $\mathbb{C}$ is identified with $\mathbb{R}^2$, this result may be rephrased to say that no restriction  of the complex exponential function to an open disc in $\mathbb{C}$ is definable in $\rexp$. Bianconi went further in \cite{bianconiundef} and showed that if $f:D\rightarrow\mathbb{C}$ is holomorphic and definable in $\rexp$ then $f$ is algebraic. This result is used by Peterzil and Starchenko in \cite{newtonpetstar} to characterise all definable locally analytic subsets of $\mathbb{C}^n$ in the structure $\rexp$.
\par Formulated in this way, the question can be generalised to other functions. There certainly are transcendental functions $f$ such that there are discs $D\subseteq\mathbb{C}$ with $f|_D$ definable in the structure $(\bar{\mathbb{R}},f|_I)$ for some interval $I\subseteq\mathbb{R}$. Indeed it turns out that examples occur with functions not so different from $\exp$. Recall that to a lattice $\Omega\subseteq \mathbb{C}$ (i.e a discrete subgroup of rank 2) Weierstrass associated a meromorphic function, $$\wp(z)=\wp_\Omega(z)=\frac{1}{z^2}+\sum_{\substack{\omega\in\Omega^*}}\left(\frac{1}{(z-\omega)^2}-\frac{1}{\omega^2}\right),$$ where $\Omega^*=\Omega\setminus\{0 \}$. This meromorphic function $\wp$ has poles at exactly the points in $\Omega$ and is periodic with respect to $\Omega$. In Section \ref{background} we shall state the addition formula for $\wp$ and its differential equation. We can see that $\wp$ is similar to $\exp$ as they are both periodic and have an addition formula and a differential equation. They are also similar as they both give an exponential map of a commutative algebraic group. \par In the course of his investigations into the model theory of these elliptic functions, Macintyre observed in \cite{macwp} that if $\Omega=\mathbb{Z}+i\mathbb{Z}$ then in the structure $(\bar{\mathbb{R}},\wp|_{(1/8,3/8)})$ the restriction of $\wp$ to any disc $D$ on which $\wp$ is analytic is definable. (The interval (1/8,3/8) is chosen for convenience as it avoids both poles of $\wp$ and the zeros of $\wp^\prime$. Any such interval may be chosen.)
\par However the lattice $\Omega=\mathbb{Z}+i\mathbb{Z}$ is rather special. It can easily be seen that $\wp(iz)=-\wp(z)$. This formula is all that is required to prove the aforementioned observation of Macintyre in \cite{macwp}. In particular there are non-integer $\alpha \in \mathbb{C}$ such that $\alpha\Omega\subseteq\Omega$. (Any $\alpha \in i\mathbb{Z}$ for example). Such lattices are said to have \textit{complex multiplication}. The lattice $\Omega=\mathbb{Z}+i\mathbb{Z}$ is also closed under complex conjugation in other words $\bar{\Omega}=\Omega$. Lattices satisfying this latter property are known as \textit{real lattices}. It can be seen in Section 18 of \cite{duval} that if $\Omega$ is a real lattice then $\wp$ is real valued when restricted to an interval on either the real or imaginary axis. Hence such restrictions of $\wp$ are real analytic functions and it is natural to consider the model theory of these restrictions. In Section \ref{mac} it is shown that Macintyre's observation extends easily to all real lattices with complex multiplication. It can then be shown that this characterises those $\Omega$ for which $\wp|_I$ for some finite real interval $I$ defines $\wp|_D$ for some disc $D$. This is done in the following theorem.
\begin{theorem}\label{theorem}
	Let $\Omega$ be a real lattice and let $\wp_{\Omega}$ be its $\wp$ function. Let $I$ be some real interval not containing a pole. Then there's a non-empty disc $D \subseteq \mathbb{C}$ such that $\wp|_D$ is definable in the structure $(\bar{\mathbb{R}},\wp|_I)$ if and only if there's an $\alpha \in \mathbb{C}\setminus\mathbb{Z}$ such that $\alpha\Omega\subseteq\Omega$.
\end{theorem}
In Section \ref{theoremproof} we give the proof of Theorem \ref{theorem}. The theorem is proved by adapting the method of Bianconi in \cite{bianconi}. In \cite{jks} Jones, Kirby and Servi attempt to apply this method to answer questions about the local interdefinability of Weierstrass $\wp$ functions. However it turns out this method cannot be applied to their problem and they turn to the method of predimensions of Hrushovski in \cite{ehstrongmin}. In fact their method cannot be used here. The method of Bianconi involves using a theorem of Wilkie on smooth functions that are defined implicitly that was proved generally by Jones and Wilkie in \cite{jw}. Bianconi refers to these methods of Wilkie as the ``Desingularisation Theorem". We shall obtain this implicit  definition in Section \ref{impdefn} and also explain why this implicit definition may be used to prove Theorem \ref{theorem}.
\section{Background on the Weierstrass $\wp$ function}\label{background}
Here we give all the background that shall be needed on the Weierstrass $\wp$ function. All of this can be found in the books \cite{kc} or \cite{silverman}. The function $\wp$ is analytic except at its poles which are at exactly the points in the lattice $\Omega$. Also $\wp$ is doubly periodic and $\Omega$ is its period lattice. Clearly $\wp$ depends on $\Omega$. As mentioned in the introduction to this paper $\wp$ has an addition formula, which is now stated. Let $z,w$ be complex numbers such that $z-w \notin \Omega$. Then, \begin{equation}
\label{wpadd}\wp(z+w)=\frac{1}{4}\left(\frac{\wp^{\prime}(z)-\wp^{\prime}(w)}{\wp(z)-\wp(w)}\right)^2-\wp(z)-\wp(w).
\end{equation}This gives rise to the duplication formula,  \begin{equation}\label{wpdup}
\wp(2z)=\frac{1}{4}\left(\frac{\wp^{\prime\prime}(z)}{\wp^{\prime}(z)}\right)^2 -2\wp(z).
\end{equation} Another important property of $\wp$ is that it satisfies a differential equation namely \begin{equation}\label{wpdiff}
(\wp^{\prime}(z))^2=4\wp^3(z)-g_2\wp(z)-g_3,
\end{equation} where the complex numbers $g_2$ and $g_3$ are called the \textit{invariants of $\wp$}. From this differential equation it is clear that there is an algebraic relation between $\wp$ and $\wp^{\prime}$. Differentiating both sides of this equation gives us that \begin{equation}\label{wp2diff}
\wp^{\prime\prime}(z)=6\wp^2(z)-\frac{g_2}{2}.
\end{equation} \par An \textit{elliptic function} $f$ is a meromorphic function on $\mathbb{C}$ which has two complex periods $\omega$ and $ \omega^{\prime}$ such that $\Im (\omega^{\prime}/\omega)>0$. These periods generate a lattice of periods $\Omega$. If we denote the field of elliptic functions for a fixed period lattice $\Omega$ by $L$ then it is known that this field $L$ is in fact $\mathbb{C}(\wp,\wp^\prime)$. This can be seen in Theorem 3.2 in Chapter 6 of \cite{silverman}.
\begin{remark}\label{reallattices}
	From Section 2 of \cite{duval} it can be seen that real lattices may be split into two cases which are known as the \textit{rectangular} and \textit{rhombic} lattices. A lattice $\Omega$ is called a \textit{rectangular lattice} if we can choose generators $\omega_1$ and $\omega_2$ of $\Omega$ such that $\omega_1$ is real and $\omega_2$ is purely imaginary. The lattice $\Omega$ is called a \textit{rhombic lattice} if we can choose generators of $\Omega$, namely $\omega_1$ and $\omega_2$ such that $\overline{\omega_1}=\omega_2$.\\ In Section \ref{theoremproof} we give the proof of Theorem \ref{theorem} in the rectangular lattice case, the proof of the rhombic case is similar.
\end{remark}
\par Finally we conclude this background section by stating a version of Ax's theorem for the Weierstrass $\wp$ function. This is a theorem of Ax in \cite{axadg1} and Brownawell and Kubota in \cite{bkpaper}.

\begin{theorem}\label{BK}
	Let $\Omega$ be a complex lattice which does not have complex multiplication. Let $z_1,\dots,z_n$ be power series with complex coefficients and no constant term that are linearly independent over $\mathbb{Q}$. Then we have that $$\trdeg_{\mathbb{C}}\mathbb{C}[z_1,\dots,z_n,\wp(z_1),\dots,\wp(z_n)]\ge n+1.$$
\end{theorem}

\section{Macintyre's Result}\label{mac}
Recall that if a complex lattice $\Omega$ has complex multiplication then there's a non-integer $\alpha$ such that $\alpha \Omega\subseteq\Omega$.
\begin{lemma}\label{maclemma}
	Let $\Omega$ be a real lattice that is closed under complex conjugation and let $\wp_\Omega$ be its $\wp$ function. Let $I$ be some real interval not containing a pole. Then the restriction of $\wp$ to any complex disc not meeting any poles is definable in the structure $(\bar{\mathbb{R}},\wp|_I)$.
\end{lemma}
\begin{proof}
	We follow the proof of Macintyre in \cite{macwp} for the case $\Omega=\mathbb{Z}+i\mathbb{Z}$. This just checks the proof works in the general case. Using the extra symmetry of the lattice $\Omega$, due to complex multiplication, we now show that $\wp$ restricted to $\alpha I$, where $\alpha\in\mathbb{C}\setminus\mathbb{Z}$ is such that $\alpha\Omega\subseteq\Omega$, is definable in the structure $(\bar{\mathbb{R}},\wp|_I)$. Let $z \in I$ and $f(z)=\wp(\alpha z).$ Then for any $\omega \in \Omega$, we can see that, $$f(z+\omega)=\wp(\alpha z+\alpha\omega)=\wp(\alpha z)=f(z)$$ as $\alpha \Omega\subseteq\Omega$. Therefore $f$ is a meromorphic function that is doubly periodic with respect to the lattice $\Omega$. So $f$ is an elliptic function for the lattice $\Omega$. Hence $f$ may be written as a rational function $R$ in $\wp(z)$ and $\wp^\prime(z)$. Similarly the function $g(z)=\wp^\prime(\alpha z)$ may be written as a rational function $S$ in $\wp(z)$ and $\wp^\prime(z)$. \par Therefore both the functions $\wp$ and $\wp^\prime$ restricted to $\alpha I$ are definable in the structure $(\bar{\mathbb{R}},\wp|_I)$. Now consider some disc $D$ contained in $I\times\alpha I$. For $z\in D$ it is clear that we may write $z=x+\alpha y$ for $x,y\in I$. We can assume that $x-\alpha y \notin \Omega.$  Then by the addition formula, $$\wp(z)=\wp(x+\alpha y)=\frac{1}{4}\left(\frac{\wp^\prime(x)-S(\wp(y),\wp^\prime(y))}{\wp(x)-R(\wp(y),\wp^\prime(y))} \right)^2-\wp(x)-R(\wp(y),\wp^\prime(y)).$$ As every function in this expression is definable we have that the function $\wp|_D$ is definable in the structure $(\bar{\mathbb{R}},\wp|_I)$. Using the addition formula gives us that $\wp$ restricted to any disc in $\mathbb{C}$ is definable in $(\bar{\mathbb{R}},\wp|_I)$ as required.
\end{proof}
\section{Getting an implicit definition}\label{impdefn}
In this section we shall describe the implicit definition that shall be needed in the proof of Theorem \ref{theorem}. This implicit definition is due to Wilkie in \cite{wilkieexp} and is referred to by Bianconi as the "Desingularisation Theorem" and was proved in a more general form by Jones and Wilkie in \cite{jw}. We let $\tilde{\mathbb{R}}=(\bar{\mathbb{R}},\mathcal{F})$ be an expansion of $\bar{\mathbb{R}}$ by a set $\mathcal{F}$ of total analytic functions in one variable, closed under differentiation that is o-minimal with a model complete theory. Desingularisation can be used to show that if the function $f:U\rightarrow \mathbb{R}$, for some $U\subseteq\mathbb{R}$, is a definable function of $\tilde{\mathbb{R}}$ then $f$ may be defined piecewise by a system of equations whose matrix of partial derivatives is non-singular. More precisely there's some interval $I^\prime \subseteq U$, some integer $n \ge 1,$ and there are certain functions $F_1,\dots,F_n:\mathbb{R}^{n+1}\rightarrow \mathbb{R}$ (see below) and functions $f_2,\dots,f_n:I^\prime\rightarrow \mathbb{R}$ such that for all $t\in I^\prime$,  $$\begin{array}{ccc}
F_1(t,f(t),f_2(t),\dots,f_n(t))=0\\\vdots\\ F_{n}(t,f(t),f_2(t),\dots,f_n(t))=0
\end{array}$$ and $$\det \left(\frac{\partial F_i}{\partial x_j}\right)_{\substack{i=1,\dots,n\\j=2,\dots,n+1}}(t,f(t),f_2(t),\dots,f_n(t))\ne 0.$$
\par Here the functions $F_1,\dots,F_n$ are polynomials in $x_1,\dots,x_{n+1}$ and $g_i(x_j)$ for $i=1,\dots,l$ and $j=1,\dots,n+1$ where $g_1,\dots,g_l\in \mathcal{F}$.  \par To apply this to our setting we can see from \eqref{wp2diff} that there is a polynomial expression for $\wp^{\prime\prime}$ in terms of $\wp$ and its invariants. Using this expression and the differential equation \eqref{wpdiff} for $\wp$ we may find similar expressions in $\wp$ and $\wp^{\prime}$ for every derivative of $\wp.$ Hence we have that the ring of terms with parameters from $\mathbb{R}$ of the structure $(\bar{\mathbb{R}},\wp|_I)$ is closed under differentiation. The structure $(\bar{\mathbb{R}},\wp|_I)$ is model complete by a theorem of Gabrielov in \cite{gab}. Bianconi also has model completeness results involving the $\wp$ function in \cite{bianconimc} however these seem difficult to apply here as they are for the complex functions rather than their restrictions to a real interval.
\par The main obstacle to using this implicit definition in the proof of Theorem \ref{theorem} is that the function $\wp|_I$ is not a total function on $\mathbb{R}.$ In order to use this method these functions are made to be total functions on $\mathbb{R}$. We do this by composing $\wp$ with a bijection from $\mathbb{R}$ to the interval $I$. This bijection is defined for a general interval $I=(a,b)$ here. In the proof of Theorem \ref{theorem} we give an explicit interval but do not explicitly define the bijection. Firstly define $A:(a,b)\rightarrow\mathbb{R}$ by $$A(t)=\frac{t-\frac{b+a}{2}}{\left(\frac{b-a}{2}\right)^2-\left(t-\frac{b+a}{2}\right)^2},$$ which is a bijection. Differentiating gives, \begin{align*}
A^\prime (t)&= \frac{\left(\frac{b-a}{2}\right)^2+\left(t-\frac{b+a}{2}\right)^2}{\left( \left(\frac{b-a}{2}\right)^2-\left(t-\frac{b+a}{2}\right)^2\right)^2},
\end{align*}which clearly doesn't vanish. The inverse, $B=A^{-1}$ is also differentiable and $$B^\prime(t)=\frac{\left(\left(\frac{b-a}{2}\right)^2-\left(B(t)-\frac{b+a}{2}\right)^2\right)^2}{\left(\frac{b-a}{2}\right)^2+\left(B(t)-\frac{b+a}{2}\right)^2}$$ also doesn't vanish.
Finally we define $$B_1(t)=\frac{1}{\left(\frac{b-a}{2}\right)^2+\left(B(t)-\frac{b+a}{2}\right)^2}.$$Therefore the structure $(\bar{\mathbb{R}},\wp\circ B,B,B_1)$ is an expansion of $\bar{\mathbb{R}}$ by total analytic functions on $\mathbb{R}$. The structures $(\bar{\mathbb{R}},\wp|_I)$ and $(\bar{\mathbb{R}},\wp\circ B,B,B_1)$ are equivalent in the sense that they have the same definable sets. Hence the structure $(\bar{\mathbb{R}},\wp\circ B,B,B_1)$ is also model complete and has a ring of terms with parameters from $\mathbb{R}$ that is closed under differentiation. Therefore in the proof of Theorem \ref{theorem} it is sufficient to pass from the structure $(\bar{\mathbb{R}},\wp|_I)$ to the auxiliary structure $(\bar{\mathbb{R}},\wp\circ B,B,B_1)$ and prove the theorem in this structure. In the next section it is explained how we may use this implicit definition in the structure $(\bar{\mathbb{R}},\wp\circ B,B,B_1)$ in order to prove Theorem \ref{theorem}.
\section{Proof of Theorem \ref{theorem}}\label{theoremproof}
In this section we give the proof of Theorem \ref{theorem}. The argument uses a method of Bianconi in \cite{bianconi} as well as \cite{bianconiundef}, which involves finding upper and lower bounds on the transcendence degree of some finite extension of $\mathbb{C}$ that are contradictory. The lower bound is found using Theorem \ref{BK} and the upper bound requires a similar argument to that used to prove Claim 4 in the proof of Theorem 4 in \cite{bianconiundef}.
\begin{proof}[Proof of Theorem \ref{theorem}.]
	By Remark \ref{reallattices} we know that real lattices may be divided into two cases, the rectangular and rhombic lattices. Firstly we note that one direction of Theorem \ref{theorem} is proved in Lemma \ref{maclemma}. Here we prove the other direction of Theorem \ref{theorem} for the rectangular lattice case as both cases are fairly similar. The main difference is the choice of generators of $\Omega$ that may be made and the intervals that can be chosen accordingly. The full details of both cases will appear in my PhD thesis.
	\par Let $\Omega$ be a rectangular real lattice. From Section 19 of \cite{duval} we can see that we can choose generators for the lattice $\Omega$, namely $\omega_1$ and $\omega_2$ such that $\omega_1$ is real and $\omega_2$ is purely imaginary. This leads to two cases, when $|\omega_1|\le |\omega_2|$ and when $|\omega_2|\le |\omega_1|$. We shall want to find a real interval $I$ not containing any lattice points such that $iI$ also doesn't contain any lattice points. In the first case the interval $(\omega_1/8,3\omega_1/8)$ is chosen and in the second case the interval $(\omega_2/8i,3\omega_2/8i)$ is chosen. It suffices to prove the theorem in either case and so we assume we are in the first of these cases and that $I=(\omega_1/8,3\omega_1/8)$. Due to the discussion in the previous section we know that it suffices to prove the theorem in the auxiliary structure $(\bar{\mathbb{R}},\wp\circ B,B,B_1)$ and we therefore now pass to this structure.

	\par Now we assume for a contradiction that $\Omega$ doesn't have complex multiplication and that there's a non-empty disc $D\subseteq\mathbb{C}$ such that the restriction $\wp|_D$ is definable in the structure $(\bar{\mathbb{R}},\wp\circ B,B,B_1)$. By translating and scaling using the addition formula \eqref{wpadd} we may take $D$ to contain the interval $iI=(i\omega_1/8,3i\omega_1/8)$. As $|\omega_1|\le |\omega_2|$ the interval $iI$ doesn't contain any lattice points. Hence the function $f_1:I\rightarrow \mathbb{R}$ defined by $f_1(t)=\wp(iB(t))$ is definable in the structure $(\bar{\mathbb{R}},\wp\circ B,B,B_1)$. By the discussion in Section \ref{impdefn} there's an open interval $I^\prime\subseteq(\omega_1/8,3\omega_1/8)$, some integer $n \ge 1,$ certain functions $F_1,\dots,F_n:\mathbb{R}^{n+1}\rightarrow\mathbb{R}$ and functions $f_2,\dots,f_n:I^\prime\rightarrow\mathbb{R}$ such that for all $t\in I^\prime$, $$\begin{array}{ccc}
	F_1(t,f_1,\dots,f_n)=0\\\vdots\\F_n(t,f_1,\dots,f_n)=0
	\end{array}$$ and $$\det\left(\frac{\partial F_i}{\partial x_j}\right)_{\substack{i=1,\dots,n\\j=2,\dots,n+1}}(t,f_1,\dots,f_n)\ne 0.$$ The functions $F_1,\dots,F_n$ are polynomials in \begin{align*}
	&x_1,\dots,x_{n+1},\wp(B(x_1)),\dots,\wp(B(x_{n+1})),\\&B(x_1),\dots,B(x_{n+1}),B_1(x_1),\dots,B_1(x_{n+1}).
	\end{align*} However as $B$ and $B_1$ are algebraic and $\wp$ and $\wp^\prime$ are algebraically dependent we may take the functions $F_1,\dots,F_n$ to be algebraic in $x_1,\dots,x_{n+1}$ and $\wp(B(x_1)),\dots,\wp(B(x_{n+1}))$ after potentially moving to a smaller interval.\par Take $n$ to be minimal such that there exists an interval $I^\prime$ and functions $F_1,\dots,F_n$ in $x_1,\dots,x_{n+1}$ and $\wp(B(x_1)),\dots,\wp(B(x_{n+1}))$ such that the above system of equations and non-singularity condition holds. As we are restricted to a small interval it is possible to shrink this interval to avoid the zeros of $\wp^\prime$ if necessary. In fact the interval may be shrunk repeatedly if needed.\par The desired contradiction will be obtained by considering upper and lower bounds on the transcendence degree of a certain finite extension of $\mathbb{C}$. These bounds will be incompatible.\par Firstly we find an lower bound on transcendence degree. In order to find this lower bound we want to apply Theorem \ref{BK}. To apply this theorem it must first be shown that $B(t)-B(0),B(f_1(t))-B(f_1(0)),\dots,B(f_n(t))-B(f_n(0))$ are linearly independent over $\mathbb{Q}$. As the lattice $\Omega$ doesn't have complex multiplication this linear independence is only required over $\mathbb{Q}.$
	\par This linear independence is shown by showing that linear dependence contradicts the minimality of $n$. This linear independence argument does not depend on whether the lattice $\Omega$ is rectangular or rhombic. If $B(t)-B(0),B(f_1(t))-B(f_1(0)),\dots,B(f_n(t))-B(f_n(0))$ are assumed to be linearly dependent over $\mathbb{Q}$ then it can be shown that $B(f_n(t))-B(f_n(0))$ can be written as a $\mathbb{Q}$-linear combination of $B(t)-B(0),B(f_1(t))-B(f_1(0)),\dots,B(f_{n-1}(t))-B(f_{n-1}(0))$. For any natural number $n$ it is possible to obtain a formula $\wp(nz)$ in terms of $\wp(z)$ and $\wp^\prime(z)$ using the addition and duplication formulas \eqref{wpadd} and $\eqref{wpdup}$. Similarly we can write $\wp(z)$ as an algebraic expression in $\wp(z/n)$ and $\wp^\prime(z/n)$.  Hence there's an algebraic expression for $\wp(z/n)$ in terms of $\wp(z)$ and $\wp^\prime(z)$. Therefore the functions $F_1,\dots,F_n$ may be rewritten as functions in $t,f_1,\dots,f_{n-1}$. This gives a new system of equations in $n$ variables. If this new system has a matrix of partial derivatives that's non-zero at $(t,f_1,\dots,f_{n-1})$ then there's a contradiction to the minimality of $n$. Hence it is assumed that all the minors of the matrix of partial derivatives for this system have determinant equal to zero and from this assumption a contradiction to the non-singularity of the original system is obtained. \par Now we observe that adding $i(B(t)-B(0))$ to the list $B(t)-B(0),B(f_1(t))-B(f_1(0)),\dots,B(f_n(t))-B(f_n(0))$ doesn't destroy this linear independence. For if it did then we could write for rational $a_0,\dots,a_{n}$ not all zero, $$i(B(t)-B(0))=a_0(B(t)-B(0))+a_1(B(f_1(t))-B(f_1(0)))+\dots+a_n(B(f_n(t))-B(f_n(0))).$$ The left hand side of this expression is non-real whereas the right hand side is real for all $t$ in the interval $I^\prime$, a contradiction. Now applying Theorem \ref{BK} to $iB(t),B(t),B(f_1(t)),\dots,B(f_n(t))$ gives that \begin{align*}
	\trdeg_{\mathbb{C}}\mathbb{C}[&iB(t),B(t),B(f_1(t)),\dots,B(f_n(t)),\wp(iB(t)),\\&\qquad\wp(B(t)),\wp(B(f_1(t))),\dots,\wp(B(f_n(t)))]\ge n+3.
	\end{align*}
	Now we obtain a upper bound on the transcendence degree of the same finite extension of $\mathbb{C}$ that is smaller than this lower bound. Recall that on the interval $I^\prime$ the function $f_1(t)=\wp(iB(t))$ is defined by a system of equations $F_1,\dots,F_n$ in the variables $x_1,\dots,x_{n+1}$, whose Jacobian is non-singular. For each $i=1,\dots,n$ the function $F_i$ can be written as $$ F_i(t,f_1,\dots,f_n)=P_i(t,f_1,\dots,f_n,\wp(B(t)),\wp(B(f_1(t))),\dots,\wp(B(f_n(t)))),$$ where $P_1,\dots,P_n$ are algebraic functions in $y_1,\dots,y_{2n+2}$. This is a system of $n$ equations in $2n+2$ variables. We check that the matrix $$\left(\frac{\partial P_i}{\partial y_j}\right)_{\substack{i=1,\dots,n\\j=2,\dots,2n+2}}(t,f_1,\dots,f_n,\wp(B(t)),\wp(B(f_1(t))),\dots,\wp(B(f_n(t))))$$ is of maximal rank $n$. This is done by a similar argument to that of Claim 4 in the proof of Theorem 4 in \cite{bianconiundef}. Differentiating $F_i$ with respect to $x_j$ gives $$\frac{\partial F_i}{\partial x_j}=\frac{\partial P_i}{\partial y_j}+B^{\prime}(x_j)\wp^{\prime}(B(x_j))\frac{\partial P_i}{\partial y_{j+n+1}}.$$ Therefore the matrix $(\partial F_i/\partial x_j)$ is given by multiplying the matrix $(\partial P_i/\partial y_j)$ by a $(2n+1)\times n$ matrix $M$.

	Here $M$ is the matrix whose first $n\times n$ block is the identity matrix followed by a row of zeros and finally an $n \times n$ diagonal matrix diag$(B^{\prime}(x_2)\wp^{\prime}(B(x_2)),\dots,B^{\prime}(x_{n+1})\wp^{\prime}(B(x_{n+1}))).$ In other words $M$ is the matrix $$M=\begin{pmatrix}
	1&\dots&0\\\vdots&\ddots&\vdots\\0&\dots&1\\0&\dots &0\\B^\prime(x_2)\wp^\prime(B(x_2))
	&\dots&0\\\vdots&\ddots&\vdots\\0&\dots&B^\prime(x_{n+1})\wp^\prime(B(x_{n+1}))
	\end{pmatrix}.$$ By the non-singularity of the system $F_1,\dots,F_n$ it's clear that the rows of the matrix $(\partial F_i/\partial x_j)$ are linearly independent over $\mathbb{R}$ and so the rows of the matrix $(\partial P_i/\partial y_j)$ are also linearly independent over $\mathbb{R}$. So this matrix is of maximal rank $n$ as required. Therefore $$ \trdeg_{\mathbb{C}}\mathbb{C}[t,f_1(t),\dots,f_n(t),\wp(B(t)),\wp(B(f_1)),\dots,\wp(B(f_n))]\le n+2.$$ However this is an upper bound on the transcendence degree on a slightly different extension of $\mathbb{C}$. But $iB(t)$ and $B(t)$ are algebraically dependent as are $B(\wp(iB(t)))$ and $\wp(iB(t))$ and $f_i$ and $B(f_i)$ for all $i=1,\dots,n$ as $B$ is algebraic. Therefore we have that \begin{align*}
	\trdeg_{\mathbb{C}}\mathbb{C}[iB(t),\wp(iB(t)),B(t)&,\wp(B(t)),B(f_1(t)),\dots,B(f_n(t)),\\&\qquad\wp(B(f_1(t))),\dots,\wp(B(f_n(t)))]\le n+2.
	\end{align*}So we have found upper and lower bounds on the transcendence degree of some finite extension of $\mathbb{C}$ that are incompatible. Hence we have a contradiction and the theorem is proved.
\end{proof}

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text:
\begin{document}
\maketitle

\begin{dedication}
\end{dedication}

\begin{abstract}   We consider the cases of the self-adjoint and skew-self-adjoint discrete Dirac systems,
obtain explicit expressions for reflection coefficients and show that rational reflection coefficients
and Weyl functions coincide.
\end{abstract}

{MSC(2010): 39A10, 39A12, 47A40}

\vspace{0.2em}

{\bf Keywords:}    Discrete self-adjoint  Dirac system, discrete skew-self-adjoint Dirac system, Weyl function,
reflection coefficient, B\"acklund-Darboux transformation.

\section{Introduction}\label{Intro}
\setcounter{equation}{0}
Discrete self-adjoint and skew-self-adjoint Dirac systems play an essential role in the study
of Toeplitz matrices (and corresponding measures), of discrete integrable nonlinear equations
(including isotropic Heisenberg magnet model) and of spectral theory of difference equations
(see, e.g., \cite{DerSi, FKKS, KaS, ALSJFA, ALSverb} and references therein). Weyl-Titchmarsh theory
of discrete systems is actively studied (see, e.g., \cite{CGe, Jo, SZ, ZeC} and various references therein).
In particular, Weyl--Titchmarsh theory of  discrete self-adjoint and skew-self-adjoint Dirac systems
was studied in \cite{FKKS, FKRS14, FKRS-LAA, KaS, RoS, SaSaR} (see also references
therein). It is known that Weyl--Titchmarsh (or simply Weyl) functions of continuous
Dirac systems  on the semi-axis are closely
related to the scattering data. Some particular results for the self-adjoint systems are contained, for instance, in \cite{BeM, GKS6}
and the general cases of continuous self-adjoint and skew-self-adjoint systems were treated in the recent paper \cite{ALS-Scat}.
The present article may be considered as the continuation of the paper \cite{ALS-Scat}, where the important discrete case
is dealt with.
We consider the cases of the self-adjoint and skew-self-adjoint discrete Dirac systems,
obtain explicit expressions for reflection coefficients and show that rational reflection coefficients
and Weyl functions coincide.

General-type
discrete self-adjoint Dirac system has the form:
\begin{equation} \label{0.1}
y_{k+1}(z)=(I_m+ \I z
j C_k)
y_k(z) \quad \left( k \in \BN_0
\right),
\end{equation}
where $\BN_0$ stands for the set of non-negative integers,  $I_m$ is the $m \times m$ identity matrix, $``\I"$ is the imaginary unit
($\I^2=-1$) and the $m \times m$ matrices $C_k$ are positive and  $j$-unitary:
\begin{equation} \label{0.2}
C_k>0, \quad C_k j C_k=j, \quad  j: = \left[
\begin{array}{cc}
I_{m_1} & 0 \\ 0 & -I_{m_2}
\end{array}
\right] \quad (m_1+m_2=m; \, \, m_1, \, m_2 > 0).
\end{equation}
We introduce the Jost solution and reflection coefficient of the system \eqref{0.1}, \eqref{0.2} in a similar to the continuous case way.
Namely, the Jost solution $\{F_k(z) \}$ ($z\in \BR$) of the Dirac system \eqref{0.1}, \eqref{0.2} is defined via its asymptotics
\begin{align} & \label{R!}
F_k(z)=\big(I_m +\I z
 j
\big)^{k}\big(I_m+o(1)\big), \quad k \to \infty.
\end{align}
The reflection coefficient $\clr(z)$ is introduced via the blocks of $F_0(z)$:
\begin{align} & \label{R1!}
\clr(z)= \begin{bmatrix} I_{m_1} & 0 \end{bmatrix} F_0(z)\begin{bmatrix} 0 \\ I_{m_2}  \end{bmatrix}\left(\begin{bmatrix} 0 & I_{m_2}  \end{bmatrix}F_0(z)
\begin{bmatrix} 0 \\ I_{m_2}  \end{bmatrix}\right)^{-1}.
\end{align}

Discrete skew-self-adjoint Dirac system (SkDDS) is given (see \cite{FKRS-LAA, KaS}) by the formula:
\begin{equation} \label{d1}
y_{k+1}(z)=\left(I_m+ \frac{\I}{ z}
C_k\right)
y_k(z),  \quad C_k=U_k^*jU_k \quad \left( k \in \BN_0
\right),
\end{equation}
where the matrices $U_k$ are unitary and $j$ is defined in \eqref{0.2}.

Direct and inverse problems (in terms of Weyl functions) were solved for systems \eqref{0.1}, \eqref{0.2} in \cite{FKRS14, RoS}
and for systems \eqref{d1} in \cite{FKKS, FKRS-LAA}. In particular,  in the case of rational Weyl matrix functions, direct and inverse
problems were solved explicitly using our GBDT version \cite{ALS94, ALS-JMAA, SaSaR} of the B\"acklund-Darboux transformation.
For various versions of B\"acklund-Darboux transformations and related
commutation methods see, for instance, \cite{Ci, D, GeT, Gu, KoSaTe, Mar, MS} and references therein.

The results of the paper imply that the procedures to recover systems from the Weyl functions enable us to recover
systems from the reflection coefficients as well. In the next section, we give some preliminary definitions and results
in order to make the paper self-sufficient. Two subsections of Section \ref{Reflect} are dedicated to
the reflection coefficients in the self-adjoint and skew-self-adjoint cases.

 In the paper, $\BN$ denotes the set of natural numbers, $\BR$ denotes the real axis, $\BC$ stands for the complex plane, and
$\BC_+$ ($\BC_-$) stands for the open upper (lower) half-plane. The spectrum of a square matrix $A$ is denoted by $\s(A)$.

\section{Preliminaries }\label{Prel}
\setcounter{equation}{0}
\subsection{Self-adjoint case}
\paragraph{1.}
Self-adjoint discrete Dirac system and stability of the explicit procedure to recover it from the Weyl function
was studied in our recent paper \cite{RoS}. We refer to \cite{RoS} for the preliminary definitions and results in this subsection.
The fundamental $m \times m$
solution $\{W_k \}$ of  \eqref{0.1}  is normalized by
\begin{align} \label{0.3}&
W_0(z)=I_m.
\end{align}
\begin{Dn} \label{defWeyl}  The Weyl function of the Dirac system \eqref{0.1}
$($which is given on the semi-axis $0\leq k < \infty$ and satisfies \eqref{0.2}$)$ is an $m_1\times m_2$ matrix  function $\vp(z)$ in the
lower half-plane, such that the following inequalities hold $:$
\begin{align} \label{0.4}&
\sum_{k=0}^\infty q(z)^k
\begin{bmatrix}
 \vp(z)^* &I_{m_2}
\end{bmatrix}
W_k(z)^*C_k W_k (z)
\begin{bmatrix}
 \vp(z) \\ I_{m_2}
\end{bmatrix}<\infty \quad (z\in \BC_-),
\\ & \label{0.5}
q(z):=(1+|z|^2)^{-1}.
\end{align}
\end{Dn}
(For the case $z \in \BC_+$, the definition of the Weyl function $\vp(z)$  of Dirac system \eqref{0.1}, \eqref{0.2}
was given in  \cite{FKRS14}.)

\paragraph{2.}  In order to consider the case of rational Weyl functions, we introduce generalized B\"acklund-Darboux transformation (GBDT)
of the discrete self-adjoint  Dirac systems. Each GBDT of the initial discrete Dirac system is determined by a triple $\{A, S_0, \Pi_0\}$ of  parameter matrices.
Here, we take a trivial initial system and
choose $n\in \BN$, two $n
\times n$
parameter matrices $A$ ($\det A \not=0$) and
$S_0>0$, and an $n
\times m$ parameter matrix $\Pi_0$ such that
\begin{equation} \label{0.6}
A S_0-S_0 A^*=\I \Pi_0 j \Pi_0^*.
\end{equation}
Define  the sequences $\{\Pi_r\}$ and $\{S_r\}$ $(r \geq 0)$  using the triple $\{A, S_0, \Pi_0\}$
and recursive
relations
\begin{align} & \label{0.7}
\Pi_{k+1}=\Pi_k+\I A^{-1}\Pi_k j \quad (k\geq 0),
\\ &  \label{0.8}
S_{k+1}=S_k+A^{-1}S_k (A^*)^{-1}+A^{-1}\Pi_k
\Pi_k^*(A^*)^{-1} \quad (k\geq 0).
\end{align}
From \eqref{0.6}--\eqref{0.8},  the validity of the matrix identity
\begin{equation} \label{0.9}
A S_{r}-S_{r} A^*=\I \Pi_{r} j \Pi_{r}^* \quad
(r \geq 0)
\end{equation}
follows by induction.  In the self-adjoint case, we introduce {\it admissible} triples $\{A, S_0, \Pi_0\}$ in the
following way.
\begin{Dn} \label{adm} The triple $\{A, S_0, \Pi_0\}$, where $\det A \not=0$, $S_0>0$ and
\eqref{0.6} holds, is called admissible.
\end{Dn}
In view of \eqref{0.8}, for the admissible triple we have $S_r>0$ $(r\geq 0)$. Thus, the
sequence (potential) $\{C_k\}$ $(k\geq 0)$ is well-defined by the equality
\begin{equation} \label{0.10}
C_k:=I_m+\Pi_k^*S_k^{-1}\Pi_k-\Pi_{k+1}^*S_{k+1}^{-1}\Pi_{k+1} .
\end{equation}
Moreover, from \cite[Theorem 2.5]{RoS} we see that the matrices $C_k$ satisfy \eqref{0.2}.
We say that the potential $\{C_k\}$ is {\it determined} by the
admissible triple. The potential determined by an admissible triple $\{A, S_0, \Pi_0\}$ is called {\it pseudo-exponential}.
We note that the notion of the pseudo-exponential (and {\it strictly pseudo-exponential}) potentials for the self-adjoint continuous
case was introduced first in \cite{GKS1} (see also \cite{GKS6}). In the discrete case, some additional requirements on the
{\it admissible and strongly admissible triples} (which determine pseudo-exponential and strictly pseudo-exponential,
respectively, potentials) appear.

All Weyl functions $\vp(z)$ are contractive in $\BC_-$
and all the potentials $\{C_k\}$, such that $\vp(z)$ (for the corresponding systems) are contractive and $\vp(-1/z)$ are
strictly proper rational, are determined by some admissible triples \cite{RoS}. Strongly admissible triples for the
self-adjoint case are considered in Subsection \ref{Refl1}.

We will need also the matrix function $w_A$,
which for each $k \geq 0$  is
a so called
transfer matrix function in Lev Sakhnovich form
\cite{SaL1, SaL3, SaSaR} and is defined by the relation
\begin{equation} \label{0.11}
w_A(k,\lambda):=I_m-\I j \Pi_k^*S_k^{-1}(A-\lambda
I_n)^{-1}\Pi_k.
\end{equation}
The fundamental
solution
$\{W_{k}\}$ of the Dirac system \eqref{0.1}
admits the representation
\begin{equation} \label{0.12}
W_{k}(z)=w_A(k,\,-1/z)\big(I_m +\I z
 j
\big)^{k}w_A(0,\,-1/z)^{-1} \quad (k \geq 0),
\end{equation}
where $w_A$ is defined in \eqref{0.11}.

Now, we partition $\Pi_k$ and write it down in the form
\begin{align}  \label{v7}  &
\Pi_k=\begin{bmatrix}(I_n+\I A^{-1})^k\vt_1 & (I_n-\I A^{-1})^k \vt_2 \end{bmatrix},
\end{align}
where $\vt_1$ and $\vt_2$ are $n\times m_1$ and $n\times m_2$, respectively, blocks of $\Pi_0$. Assume further in this subsection that
\begin{align}  \label{R0}  &
\pm \I \not\in \s(A).
\end{align}
In view of \eqref{0.9} and \eqref{v7}, setting
\begin{align}  \label{v8}  &
R_{r}:=(I_n+\I A^{-1})^{-r}S_r\big(I_n-\I (A^*)^{-1}\big)^{-r}
\end{align}
we have
\begin{align} \nn
R_{k+1}-R_k=& 2(I_n+\I A^{-1})^{-k-1}A^{-1}(I_n-\I A^{-1})^k \vt_2\vt_2^* \big((I_n-\I A^{-1})^k\big)^*\big(A^{-1}\big)^*
\\  \label{v9}  &
\times
\big((I_n+\I A^{-1})^{-k-1}\big)^*\geq 0.
\end{align}
Since $R_0=S_0>0$, relations \eqref{v9} imply that there is a
limit
\begin{align}  \label{v10}  &
\lim_{k\to \infty}R_k^{-1}=\vk_R\geq 0.
\end{align}
In a similar way we introduce the matrices
\begin{align}  \label{v8'}  &
Q_{r}:=(I_n-\I A^{-1})^{-r}S_r\big(I_n+\I (A^*)^{-1}\big)^{-r} ,
\end{align}
and show that
\begin{align} \label{v9'}  &
Q_{k+1}-Q_k \geq 0.
\end{align}
Since $Q_0=S_0>0$, relations \eqref{v9'} imply that there is a
limit
\begin{align}  \label{v10'}  &
\lim_{k\to \infty}Q_k^{-1}=\vk_Q\geq 0.
\end{align}

\subsection{Skew-self-adjoint case}
The preliminary definitions and results on the skew-self-adjoint discrete  Dirac systems (SkDDS) \eqref{d1}
we take from \cite{FKKS} and sometimes from \cite{FKRS-LAA}.
\begin{Rk}\label{RkNn}
The notations here slightly differ
from the notations in \cite{FKKS, FKRS-LAA}. In particular, we introduce the matrices $R_k$ and $Q_k$
in the both self-adjoint and skew-self-adjoint cases via formulas \eqref{v8} and \eqref{v8'}, respectively,
but in \cite{FKRS-LAA} $R_k$ stands for $Q_k$ in the the present notations and $Q_k$ stands for $R_k$.
\end{Rk}
\begin{Dn}\label{DnD0} The Weyl function of SkDDS is an $m_1 \times m_2$ matrix function $\vp(z)$ in
$$\BC_M=\{z\in \BC: \, \Im(z)>M\}  \quad {\mathrm{for \,\, some}} \quad  M>0,$$
which satisfies the inequality
\begin{equation} \label{k1}
\sum_{k=0}^{\infty}\begin{bmatrix}  \vp(z)^* & I_{m_2}\end{bmatrix}w_k(z)^* w_k(z)
\left[
\begin{array}{c}
 \vp(z) \\ I_{m_2}
\end{array}
\right] < \infty,
\end{equation}
where $w_k(z)$ is the {fundamental solution} of SkDDS normalized by $w_0(z) \equiv I_m$.
\end{Dn}
Let us fix again an integer $n>0$, and consider an $n \times n$ matrix $A$
with $\det \, A \not= 0$, an  $n \times n$ matrix $S_0>0$  and  an $n \times m$  matrix $\Pi_0$. These
matrices should satisfy the  identity
\begin{equation} \label{k2}
A S_0-S_0 A^*= \I \Pi_0 \Pi_0^*.
\end{equation}
The sequences $\{\Pi_k\}$, $\{S_k\}$ and $\{C_k\}$ $(k \geq 0)$ are introduced using the triple $\{A, \, S_0, \, \Pi_0\}$ and relations
\begin{align}\label{k3}
&\Pi_{k+1}= \Pi_k+\I A^{-1} \Pi_k j, \\
&
S_{k+1}=S_k+ A^{-1} S_k (A^*)^{-1}+ A^{-1} \Pi_k j \Pi_k^*
(A^*)^{-1},\label{k4}
\\ \label{k5} &
C_k=j+ \Pi_k^* S_k^{-1} \Pi_k - \Pi_{k+1}^* S_{k+1}^{-1} \Pi_{k+1}.
\end{align}
Similar to the self-adjoint case we write down $\Pi_k$ in the form
$$\Pi_k=\begin{bmatrix}(I_n+\I A^{-1})^k\vt_1 & (I_n-\I A^{-1})^k \vt_2 \end{bmatrix}.$$
If $S_0>0$,  the identity \eqref{k2} holds and the  pair $\{A, \, \vt_1\}$
is controllable, then according to \cite[Lemma 3.2]{FKKS} and   \cite[Proposition 3.6]{FKKS} we have
$\det \, A \not= 0$, $S_k>0$ and the matrices $C_k$ admit representation $C_k=U_k^*jU_k$
from \eqref{d1}. That is, the sequence $\{C_k\}$ is well-defined and the corresponding
system is a skew-self-adjoint Dirac system. In the skew-self-adjoint case, the triple $\{A, \, S_0, \, \Pi_0\}$,
such that $S_0>0$,  the identity \eqref{k2} holds and the  pair $\{A, \, \vt_1\}$
is controllable, is called {\it admissible}. The potential determined by this triple
is called {\it pseudo-exponential}.

Moreover, if $\vp(z)$ is a strictly proper rational $m_1\times m_2$ matrix function then
it is the Weyl function of some skew-self-adjoint Dirac system with the pseudo-exponential potential
(see \cite[Theorem 4.2]{FKRS-LAA}). We will require additionally that $\I \not\in \s(A)$.
\begin{Dn} In the skew-self-adjoint case, the triple $\{A, \, S_0, \, \Pi_0\}$, where  $S_0>0$, the identity \eqref{k2} is valid, the pair $\{A, \, \vt_1\}$
is controllable and $\I \not\in \s(A)$, is called strongly admissible. The potentials determined by the strongly admissible triples are called strictly pseudo-exponential.
\end{Dn}
Note that
\cite[Proposition 4.8]{FKRS-LAA} implies that if $S_0>0$,  \eqref{k2} holds and $0,\, \I \not\in \s(A)$ then $S_k>0$, the matrices $C_k$
are well-defined and there is a strongly admissible triple which determines the same potential $\{C_k\}$ as $\{A, \, S_0, \, \Pi_0\}$.
The fundamental solution $w_k$ of SkDDS determined by the  strongly admissible triple $\{A, \, S_0, \, \Pi_0\}$ has the form
\begin{align}& \label{k6}
w_k( z )=w_{ A }(k, -z ) \left(
I_{m}+\frac{\I}{z}j \right)^{k} w_{ A }(0, - z
)^{-1},
\end{align}
whereas $w_A$ in the skew-self-adjoint case is given by
\begin{align}
 & \label{k7}
w_{ A }(k, \la ):=I_{m}- \I \Pi_k^{*}
S_k^{-1} ( A- \lambda I_{n} )^{-1} \Pi_k.
\end{align}
Taking into account Remark \ref{RkNn}, we see that \cite[Proposition 4.10]{FKRS-LAA} and   \cite[(4.34)]{FKRS-LAA} imply that
\begin{align}  \label{k8}  &
\lim_{k\to \infty}Q_k^{-1}=0; \quad \lim_{k\to \infty}\big(Q_k^{-1}\wt G(A)^k\vt_1\big)=0, \quad \wt G(A):=(A-\I I_n)^{-1}(A+\I I_n)
\end{align}
in the case of a strongly admissible triple $\{A, \, S_0, \, \Pi_0\}$.

\section{Reflection coefficients}\label{Reflect}
\setcounter{equation}{0}
\subsection{Reflection coefficients: self-adjoint case }\label{Refl1}
In this subsection, we express (via the triple   $\{A, \, S_0, \, \Pi_0\}$)
the Jost solution and reflection coefficient, which are the analogs of the corresponding functions in the continuous case.

Uniqueness of the solution of the inverse problem to recover system from the Weyl function (see \cite[Theorem 2.3]{RoS}) together
with Theorems 2.6 and 2.8 and Proposition 2.7 (all from \cite{RoS}) imply that without loss of generality
one can require that $\s(A )\subset (\BC_+\cup \BR)$. Further we use a stronger requirement
\begin{align}  \label{R1}  &
\s(A )\subset \BC_+, \quad  \I \not\in \s(A).
\end{align}
Following \cite{FKKS, GKS1}, we call the admissible triple satisfying \eqref{R1} {\it strongly admissible}
and  we introduce the class of the strictly pseudo-exponential potentials
$\{C_k\}$.
\begin{Dn}\label{sps} The potentials $\{C_k\}$ of the Dirac systems \eqref{0.1}, \eqref{0.2},
which are determined by the strongly admissible triples, are called strictly pseudo-exponential.
\end{Dn}
In view of \eqref{0.11}, \eqref{v7}, \eqref{v8} and \eqref{v8'}, we have a representation
\begin{align}\label{R2}
& w_{A}(k,-1/ z )
\\ \nn &=
I_m-\I zj
\begin{bmatrix}
\vt_1^*R_k^{-1}(I_n+zA)^{-1}\vt_1 &
\vt_1^*R_k^{-1}(I_n+zA)^{-1} G(A)^k\vt_2
\\
\vt_2^*(G(A)^k)^*R_k^{-1} (I_n+zA)^{-1}\vt_1 &
\vt_2^*Q_k^{-1}(I_n+zA )^{-1}\vt_2
\end{bmatrix},
\end{align}
where
\begin{align}  \label{R3}  &
G(A)=(I_n+\I A^{-1})^{-1}(I_n-\I A^{-1}).
\end{align}
Relations \eqref{R1} and \eqref{R3} yield
\begin{align}  \label{R4}  &
\s\big(G(A)\big) \subset \{\la: \, |\la |<1 \}.
\end{align}
Hence, from \eqref{v10}, \eqref{v10'} and \eqref{R2} we derive
\begin{align}\label{R5}
& \lim_{k\to \infty} w_{A}(k,-1/ z )
=
\begin{bmatrix}
\chi_1(z) & 0
\\
0 & \chi_2(z)
\end{bmatrix},
\\ \label{R6} &
 \chi_1(z):=I_{m_1}-\I z \vt_1^*\vk_R (I_n+zA)^{-1}\vt_1, \,\, \chi_2(z):=I_{m_2}+\I z \vt_2^*\vk_Q (I_n+zA)^{-1}\vt_2.
\end{align}
According to \eqref{R!}, \eqref{0.12} and \eqref{R5}, the Jost solution $\{F_k\}$ is given by the equalities
\begin{align}  \label{R7}  &
F_k(z)=W_k(z)w_{A}(0,-1/ z )\begin{bmatrix}
\chi_1(z)^{-1} & 0
\\
0 & \chi_2(z)^{-1}
\end{bmatrix}.
\end{align}
Partition $w_A$ into the blocks corresponding to the partitioning of $j$:
\begin{align}\label{R8}&
 w_A(0,\lambda)=\left[
\begin{array}{lr}
a(\lambda) & b(\lambda) \\ c(\lambda) & d(\lambda)
\end{array}
\right].
\end{align}
It was shown in the proof of \cite[Theorem 2.6]{RoS} (see \cite[(2.29)]{RoS}) that the Weyl function $\vp(z)$
of the system \eqref{0.1}, \eqref{0.2} (in $\BC_-$)
is given by the formula
\begin{align}\label{R9}&
\vp(z)=b(-1/z)d(-1/z)^{-1}.
 \end{align}
Relations  \eqref{R1!}, \eqref{0.3} and \eqref{R7}--\eqref{R9} imply the following theorem.
\begin{Tm} \label{Tm1} Let Dirac system \eqref{0.1}, \eqref{0.2}
be a system with the
strictly pseudo-exponential potential $\{C_k\}$. Then the
Weyl function $\vp(z)$ is the unique analytic continuation of the
reflection coefficient $\clr(z)$ of this system. That is, the reflection
coefficient and the Weyl function are given by the same rational matrix function.
\end{Tm}
From \cite[Theorem 2.6]{RoS} and Theorem \ref{Tm1} we derive the following corollary.
\begin{Cy}\label{Cy2} Let the potential $\{C_k\}$ be determined by a strongly admissible triple $\{A, \, S_0, \, \Pi_0\}$.
Then, the reflection coefficient of the Dirac system \eqref{0.1}, \eqref{0.2} is given by the formula
\begin{equation} \label{R10}
\clr(z)=-\I z\vt_1^*S_0^{-1}(I_n+zA^{\times})^{-1}\vt_2,
\quad A^{\times}=A+\I \vt_2 \vt_2^*S_0^{-1}.
\end{equation}
\end{Cy}
\subsection{Reflection coefficients: skew-self-adjoint case }\label{Refl2}
In the skew-self-adjoint case, we define the reflection coefficient $\clr(z)$ in a slightly more general way than in
the self-adjoint case.  That is, we consider the matrix valued  $m_2 \times m$ solution $Y$ of the system \eqref{d1}:
\begin{equation} \label{R11}
Y_k(z)=\big(1-\frac{\I}{z}\big)^k\left(\begin{bmatrix} 0 \\ I_{m_2}\end{bmatrix}+o(1)\right), \quad k\to \infty ,
\end{equation}
and set
\begin{align} & \label{R12}
\clr(z)= \begin{bmatrix} I_{m_1} & 0 \end{bmatrix} Y_0(z)\left(\begin{bmatrix} 0 & I_{m_2}  \end{bmatrix}Y_0(z)\right)^{-1}.
\end{align}

In order to express $\clr(z)$ via a strongly admissible  triple $\{A, \, S_0, \, \Pi_0\}$, we derive from \eqref{v7}, \eqref{v8}, \eqref{v8'} and
\eqref{k7} the representation
\begin{align}& \label{R13}
 w_{A}(k,-z )\begin{bmatrix} 0 \\ I_{m_2}\end{bmatrix}=
\begin{bmatrix} 0 \\ I_{m_2}\end{bmatrix}
-\I
\begin{bmatrix}
\vt_1^*\big(\wt G(A)^k\big)^*Q_k^{-1}(z I_n+A)^{-1}\vt_2
\\
\vt_2^*Q_k^{-1}(z I_n+A )^{-1}\vt_2
\end{bmatrix},
\end{align}
where $\wt G$ is introduced in \eqref{k8}. Formulas \eqref{k8} and \eqref{R13} imply that
\begin{align}& \label{R14}
\lim_{k \to \infty} w_{A}(k,-z )\begin{bmatrix} 0 \\ I_{m_2}\end{bmatrix}=\begin{bmatrix} 0 \\ I_{m_2}\end{bmatrix}.
\end{align}
It follows from \eqref{k6}, \eqref{R11} and \eqref{R14} that
\begin{align}& \label{R15}
 Y_k(z)=\big(1-\frac{\I}{z}\big)^k w_{A}(k,-z )\begin{bmatrix} 0 \\ I_{m_2}\end{bmatrix}.
 \end{align}
Hence, after we take into account \eqref{R12} and (similar to the self-adjoint case) partition $w_A$ (as in  \eqref{R8}), we obtain
\begin{align}& \label{R16}
 \clr(z)=b(-z)d(-z)^{-1}.
 \end{align}
On the other hand, according to \cite[(3.24)]{FKKS} the Weyl function $\vp(z)$ of the system \eqref{d1}
is also given by the right-hand side of \eqref{R16}.
Thus, the following theorem is proved.
\begin{Tm} \label{Tm2} Let Dirac system \eqref{d1}
be a system with the
strictly pseudo-exponential potential $\{C_k\}$. Then the
Weyl function $\vp(z)$ is the analytic continuation of the
reflection coefficient $\clr(z)$ of this system. More precisely, the reflection
coefficient and the Weyl function are given by the same rational matrix function.
\end{Tm}
The next corollary follows from \cite[Theorem 3.8]{FKKS} and Theorem \ref{Tm2}.
\begin{Cy}\label{Cy5} Let the potential $\{C_k\}$ be determined by a strongly admissible triple $\{A, \, S_0, \, \Pi_0\}$.
Then, the reflection coefficient of the skew-self-adjoint Dirac system \eqref{d1} is given by the formula
\begin{equation} \label{R17}
\clr(z)=-\I \vt_1^*S_0^{-1}(zI_n+zA^{\times})^{-1}\vt_2,
\quad A^{\times}=A-\I \vt_2 \vt_2^*S_0^{-1}.
\end{equation}
\end{Cy}

\bigskip

\noindent{\bf Acknowledgments.}
 {The research   of  A.L. Sakhnovich was supported by the
Austrian Science Fund (FWF) under Grant  No. P29177.}


text:
\section{Equivalence of Definitions of Ordering/Proof 2}
Tags: Equivalence of Definitions of Ordering

\begin{theorem}
The following definitions of [[Definition:Ordering|ordering]] are equivalent:
\end{theorem}

\begin{proof}
=== [[Definition:Ordering/Definition 1|Definition 1]] implies [[Definition:Ordering/Definition 2|Definition 2]] ===
Let $\RR$ be a [[Definition:Relation|relation]] on $S$ satisfying:
{{begin-axiom}}
{{axiom | n = 1
        | lc= $\RR$ is [[Definition:Reflexive Relation|reflexive]]
        | q = \forall a \in S
        | m = a \mathrel \RR a
}}
{{axiom | n = 2
        | lc= $\RR$ is [[Definition:Transitive Relation|transitive]]
        | q = \forall a, b, c \in S
        | m = a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c
}}
{{axiom | n = 3
        | lc= $\RR$ is [[Definition:Antisymmetric Relation|antisymmetric]]
        | q = \forall a \in S
        | m = a \mathrel \RR b \land b \mathrel \RR a \implies a = b
}}
{{end-axiom}}
==== Condition $(1)$ ====
Let $\tuple {x, y} \in \RR \circ \RR$.
Then there exists a $z \in \RR$ such that:
:$\tuple {x, z}, \tuple {z, y} \in \RR$
By $\RR$ being [[Definition:Transitive Relation|transitive]]:
:$\tuple {x, y} \in \RR$
Hence:
:$\RR \circ \RR \subseteq \RR$
Now let $\tuple {x, y} \in \RR$.
By $\RR$ being [[Definition:Reflexivity|reflexive]]:
:$\tuple {y, y} \in \RR$
Hence by the definition of [[Definition:Composition of Relations|relation composition]]:
:$\tuple {x, y} \in \RR \circ \RR$
Hence:
:$\RR \subseteq \RR \circ \RR$
==== Condition $(2)$ ====
Follows immediately from [[Relation is Antisymmetric iff Intersection with Inverse is Coreflexive]] and $\RR$ being [[Definition:Reflexive Relation|reflexive]].
Thus $\RR$ is an [[Definition:Ordering/Definition 2|ordering by definition 2]].
{{qed|lemma}}
=== [[Definition:Ordering/Definition 2|Definition 2]] implies [[Definition:Ordering/Definition 1|Definition 1]] ===
Let $\RR$ be a [[Definition:Relation|relation]] which fulfils the conditions:
:$(1): \quad \RR \circ \RR = \RR$
:$(2): \quad \RR \cap \RR^{-1} = \Delta_S$
==== Reflexivity ====
By [[Intersection is Subset]] the condition:
:$\RR \cap \RR^{-1} = \Delta_S$
implies:
:$\Delta_S \subseteq \RR$
Thus $\RR$ is [[Definition:Reflexive Relation/Definition 2|reflexive]] by definition.
==== Antisymmetry ====
By [[Relation is Antisymmetric iff Intersection with Inverse is Coreflexive]] the condition:
:$\RR \cap \RR^{-1} = \Delta_S$
implies that $\RR$ is [[Definition:Antisymmetric Relation|antisymmetric]].
==== Transitivity ====
Let $\tuple {x, y}, \tuple {y, z} \in \RR$.
Then by the definition of [[Definition:Composition of Relations|relation composition]]:
:$\tuple {x, z} \in \RR \circ \RR$
But by the condition:
:$\RR \circ \RR = \RR$
It follows that:
:$\tuple {x, z} \in \RR$
Hence $\RR$ is [[Definition:Transitive Relation|transitive]].
Thus $\RR$ is an [[Definition:Ordering/Definition 1|ordering by definition 1]].
{{qed}}
\end{proof}

config: wiki


TITLE: Conditions for the existence of a lower bound of the operator norm $\|A^*A\|$ for a linear and continuous operator $A:X \to Y$
QUESTION [0 upvotes]: Setting: $X$, $Y$ Banach (or Hilbert) spaces, $A: X \to Y$ linear, continuous, injective, $A(X)$ is dense in $Y$ ($A$ is an imbedding operator in a Gelfand triple), $A^*$ is its adjoint

Find: Conditions on $A$ for the existence of a constant $C>0$ such that $\|A^*A\|=\sup_x \frac{\|A^*Ax\|_Y}{\|x\|_X} \geq C$.

You are free to propose additional assumptions, e.g. compactness of $A$.

I imagine you could use a spectral theorem to find a lower bound with
eigenvalues, afaik it was $\|A^*A\|\geq |\lambda|$ for $\lambda \in
   \sigma(A^*A)$. Or the optimal lower bound is the spectral radius. I do not really want to assume self-adjointness.

Of course, unitary operators (i.e. $A^*A=Id$) are a class of operators with this feature.

I don't know if it helps, but since $A(X)$ is dense in $Y$, we know
that $A^*$ and $A^*A$ are injective

REPLY [1 votes]: If $C$ can depend on $A$ we can take $C=\|A^{*}A\|$. You cannot have $C$ independent of $A$. For example take $K$ to be an injective compact operator with dense range and let $A_n =\frac 1 n K$. Your  inequality obviously fails for large $n$.


TITLE: About quadratic map
QUESTION [0 upvotes]: let us consider the following quadratic map:
$$s_{n}=s_{n-1}²+c$$ $$(*)$$
There is several papers disscuting the dynamics of (*). I want to know the behavior of this map for $c=-2$ and I am asking if this map is equivalent to the logistic map $$s_{n}=rs_{n-1}(1-s_{n-1})$$ for some $0≤r≤4$.

REPLY [1 votes]: $rs_n=rs_{n-1}(r-rs_{n-1})$, let $t_n=-rs_n$,
so $t_n+r/2=(t_{n-1}+r/2)^2-r^2/4+r/2$, let $u_n=t_n+r/2$
and $u_n=u_{n-1}^2-(r^2-2r)/4$.
So $r^2-2r=8$,$(r-1)^2=9$,$r=4$
$u_n=2-4s_n$, $u_n=u_{n-1}^2-2$,$s_n=4s_{n-1}(1-s_{n-1})$


TITLE: What is the angle in a complex inner product?
QUESTION [1 upvotes]: The dot product in 2D or 3D Euclidean space between vectors $\mathbf{a}$ and $\mathbf{b}$ with magnitudes $|\mathbf{a}|$ and $|\mathbf{b}|$ can be written as
$$
\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|}
$$
where $\theta$ is the angle between the vectors. So far so good.
Now using vectors in $\mathbb{C}^2$, we can find the inner product $\langle \mathbf{a},\mathbf{b} \rangle$ by finding the complex conjugate of $\mathbf{a}$ to get $\overline{\mathbf{a}}$ and multiplying that with $\mathbf{b}$ (I understand mathematicians conjugate the second term), giving us
$$
\langle\mathbf{a},\mathbf{b}\rangle = \sum_{i=0}^{n-1} \overline{a_i}\,b_i
$$
If we think of this as a generalization of the dot product, then we can think the above as also $|\overline{\mathbf{a}}|\,|\mathbf{b}|\,\cos\theta$ for the angle $\theta$ between $\overline{\mathbf{a}}$ and $\mathbf{b}$. This is not the angle between $\mathbf{a}$ and $\mathbf{b}$ because $\overline{\mathbf{a}}$ is not a scaled version of $\mathbf{a}$ - it points in an entirely different direction.
I find this confusing! We're not really generalizing one of the most useful properties of the dot product because we're not getting back something that can give us the angle between the two vectors. We're getting back a different angle, involving $\overline{\mathbf{a}}$ rather than $\mathbf{a}$. Yet I frequently see the inner product used as a generalization of the dot product in this respect.
Can anyone please straighten me out? Definitions are less useful than explanations, as I want to know where this line of thinking is in error.

REPLY [0 votes]: To get inner product of vectors a and b we need to take conjugate transpose of a and then multiply it to vector b. But taking conjugate transpose of a is just an intermediate step; it does not mean that we get inner product between conjugate transpose of a and b. What we get is actually the inner product of a and b. So the angle defined by the inner product is the angle between vectors a and b.