text:
\section{#1}}

\newcommand{\refroyalsub}[2]{\ref{#2}}
\newcommand{\refappparen}[1]{(Appendix~\ref{#1})}

\newcommand{\mtilde}[1]{#1}

\newcommand{k}{k}

\begin{document}

\title[The Minimum Number of Rotations About Two Axes]{The Minimum Number of Rotations About Two Axes for Constructing an Arbitrarily Fixed Rotation}

\author{%
Mitsuru Hamada}
\iffalse
\address{%
Quantum Information Science Research Center,
Quantum ICT Research Institute,
   Tamagawa University,
Tamagawa-gakuen 1-chome, Machida, Tokyo 194-8610, Japan}
\fi
\thanks{%
The author is with
Tamagawa University,
Tamagawa-gakuen 1-chome, Machida, Tokyo 194-8610, Japan.}

\iffalse
\curraddr{}
\email{}
\thanks{}
\fi

\keywords{SU(2), SO(3), rotation}

\date{}

\dedicatory{}

\begin{abstract}
For any pair of three-dimensional real unit vectors $\myv{m}$ and $\myv{\varvn}$ with $\abs{\rinnpr{\myv{m}}{\myv{\varvn}}} < 1$ and
any rotation $U$, let $N_{\myv{m},\myv{n}}(U)$ denote
the least value of a positive integer $k$ such that $U$
can be decomposed into
a product of $k$ rotations about either $\myv{m}$ or $\myv{\varvn}$.
This work gives the number $N_{\myv{m},\myv{n}}(U)$
as a function of $U$.
Here a rotation means an element $D$ of the special orthogonal group ${\rm SO}(3)$
or an element of the special unitary group ${\rm SU}(2)$ that corresponds to $D$.
Decompositions
of $U$ attaining the minimum number $N_{\myv{m},\myv{n}}(U)$
are also given explicitly.
\end{abstract}

\maketitle

\section{Introduction \label{ss:intro}}

In this work,
an issue on optimal constructions of rotations in the
Euclidean space $\mymathbb{R}^3$, under some restriction,
is addressed and solved.
By a rotation or rotation matrix, we usually mean an element of the special orthogonal group ${\rm SO}(3)$.
However, we follow the custom, in quantum physics, to call not only an element of ${\rm SO}(3)$ but also that of the special unitary group ${\rm SU}(2)$
a rotation.
\begin{comment}
\footnote{Recall that ${\rm SU}(2)$ and ${\rm SO}(3)$ denote the set of $2\times 2$ unitary matrices with determinant $1$ and
the set of $3\times 3$ real orthogonal matrices with determinant $1$, respectively.}
\end{comment}
This is justified by the well-known homomorphism from ${\rm SU}(2)$
onto ${\rm SO}(3)$ (Section~\refroyalsub{ss:prel}{sssub:homomorphism}).
Given a pair of three-dimensional real unit vectors $\myv{m}$ and $\myv{\varvn}$ with $\abs{\rinnpr{\myv{m}}{\myv{\varvn}}} < 1$,
where $\myv{m}^{\rm T}$ denotes the transpose of $\myv{m}$,
let $N_{\myv{m},\myv{n}}(\cD)$ denote
the least value of a positive integer $k$ such that any
rotation in $\cD$ can be decomposed into (constructed as) a product of $k$ rotations about either $\myv{m}$ or $\myv{\varvn}$,
where $\cD = {\rm SU}(2),{\rm SO}(3)$.
It is known that
$
N_{\myv{m},\myv{n}}\big({\rm SO}(3)\big) = N_{\myv{m},\myv{n}}\big({\rm SU}(2)\big) =
\lceil  \pi / \arccos \abs{\rinnpr{\myv{m}}{\myv{\varvn}}} \rceil +1
$
for any pair of three-dimensional real unit vectors $\myv{m}$ and $\myv{\varvn}$ with $\abs{\rinnpr{\myv{m}}{\myv{\varvn}}} < 1$~\cite{Lowenthal71,Lowenthal72}.

Then, a natural question arises:
What is the least value, $N_{\myv{m},\myv{n}}(U)$, of a positive integer $k$ such that an arbitrarily fixed rotation $U$
can be decomposed into
a product of $k$ rotations about either $\myv{m}$ or $\myv{\varvn}$?
In this work,
the minimum number $N_{\myv{m},\myv{n}}(U)$ is given as an explicit
function of $U$, where $U$ is expressed in terms of
parameters
known as Euler angles~\cite{Wigner,BiedenharnLouck}.
Moreover,
optimal, i.e., minimum-achieving decompositions (constructions) of any fixed element $U \in {\rm SU}(2)$
are presented explicitly.
\begin{comment}
Naturally, the above formula on $N_{\myv{m},\myv{n}}:= N_{\myv{m},\myv{n}}\big({\rm SU}(2)\big)$ follows from the obtained stronger formula on $N_{\myv{m},\myv{n}}(U)$, as will be shown.
\end{comment}

In this work, not only explicit constructions
but also
simple inequalities on geometric quantities, which directly show lower bounds on the number of constituent rotations, will be presented.
Remarkably,
the proposed explicit constructions meet the obtained lower bounds, which shows both the optimality of the constructions and the tightness of the bounds.

The results in this work were obtained before the author came to know
Lowenthal's formula on
$N_{\myv{m},\myv{n}}\big({\rm SO}(3)\big)$~\cite{Lowenthal71,Lowenthal72} and
a related result~\cite{DA04}. Prior to the present work, the work~\cite{DA04} has treated the issue of
determining $N_{\myv{m},\myv{n}}(D)$, $D \in {\rm SO}(3)$.
The interesting result~\cite{DA04}, however, gave
$N_{\myv{m},\myv{n}}(D)$, $D \in {\rm SO}(3)$,
only algorithmically (with the largest index of a sequence of real numbers
with some property).
\iffalse
This work's expression and derivation of
$N_{\myv{m},\myv{n}}(D)$ are considerably different from those of \cite{DA04}.
\fi
The distinctive features of the present work include the following:
$N_{\myv{m},\myv{n}}(U)$ is given in terms of an explicit function of parameters of $U\in{\rm SU}(2)$;
explicit optimal decompositions are presented;
this work's results on $N_{\myv{m},\myv{n}}(U)$ imply Lowenthal's formula on $N_{\myv{m},\myv{n}}\big({\rm SO}(3)\big)$ in a consistent self-contained manner.%
\footnote{Here the crux of the difficulty in obtaining this work's results will be explained.
Finding the minimum {\em odd}\/ number of factors needed for decomposing $U$, which is expressed with a standard parameter $\beta} %{\gamma$ of $U$, together with minimum-achieving decompositions, was relatively easy.
The crux
lay in obtaining a solution to attain the minimum {\em even}\/ number of factors, which was found to be expressed with a new parameter $\beta} %{\gamma'$ eventually.}

Regarding another direction of related research, we remark that $N_{\myv{m},\myv{n}}(\cD)$ is known as the order of (uniform) generation of the Lie group $\cD$, and this notion has been extended to other Lie groups.
The interested reader is referred to relatively extensive treatments on uniform
generation~\cite{KLowenthal75,Leite91},
where one would find that
even determining the order $N_{\myv{m},\myv{n}}\big({\rm SO}(3)\big)$ needs a special proof~\cite{Lowenthal71,Lowenthal72}, \cite[Appendix]{Leite91}.

Detailed elementary arguments below would help us dispel some confusions related to $N_{\myv{m},\myv{n}}\big({\rm SU}(2)\big)$
often found in textbooks on quantum computation.
There, not to mention the ignorance of the fact
$N_{\myv{m},\myv{n}}\big({\rm SU}(2)\big)=\lceil  \pi / \arccos \abs{\rinnpr{\myv{m}}{\myv{\varvn}}} \rceil +1$, a wrong statement equivalent to saying that
$N_{\myv{m},\myv{n}}\big({\rm SU}(2)\big)$ were {\em three}, regardless of the choice of non-parallel vectors $\myv{m}$ and $\myv{n}$, is observed.

Regarding physics,
this work has been affected by the issue of constructing an arbitrary unitary operator on
a
Hilbert space
discussed in quantum physics~\cite{ReckZBB94}. This is relevant to universal gates for quantum computation~\cite{BoykinMPRV99}.
In this context,
requiring the availability of rotations about a pair of exactly orthogonal axes seems too idealistic.
For example, consider a Hamiltonian $H$ of a quantum system represented by $\mymathbb{C}^2$, and note that $H$ determines the axis of the rotations
$[c(t)]^{-1} \exp (-{i} %{{\rm i}}t H ) \in {\rm SU}(2)$, $t\in \mymathbb{R}$, where $c(t)$ is a square root of $\det \exp (-{i} %{{\rm i}}t H)$. [Often, although not always, differences of unitary matrices (evolutions) up to scalar multiples are ignorable.]
Thus, explicit decompositions attaining the minimum $N_{\myv{m},\myv{n}}(U)$ of an arbitrary rotation $U$ for the generic vectors $\myv{m}$ and $\myv{n}$ will be useful.
For applications to control, the reader is referred to \cite{DA04} and references therein.

This paper is organised as follows.
After giving preliminaries in Section~\ref{ss:prel}, the main theorem establishing $N_{\myv{m},\myv{n}}(U)$ and explicit
constructions of rotations are presented in Section~\ref{ss:const}.
Then, inequalities that show limits on constructions are presented in Section~\ref{ss:limits}.
The proofs of the results of this work are presented in Section~\ref{ss:Proofs}.
Section~\ref{ss:conc} contains the conclusion.
Several arguments are relegated to appendices.
\iffalse
 including the elementary
proof of the formula on $N_{\myv{m},\myv{n}}$~\cite{Lowenthal71,Lowenthal72}.
\fi

\section{Preliminaries and a Known Result \label{ss:prel}}

\subsection{Definitions}

The notation to be used includes the following:
$\mymathbb{N}$ denotes the set of strictly positive integers;
$S^2= \{ \myv{v} %\underline{v}} %{n} \in \mymathbb{R}^3 \mid \| \myv{v} %\underline{v}} %{n} \| =1 \}$ where
$\| \myv{v} %\underline{v}} %{n} \| = \sqrt{v} %\underline{v}} %{n_x^2+v} %\underline{v}} %{n_y^2+v} %\underline{v}} %{n_z^2}$ for $\myv{v} %\underline{v}} %{n}=(v} %\underline{v}} %{n_x,v} %\underline{v}} %{n_y,v} %\underline{v}} %{n_z)^{\rm T}$;
$\lceil x \rceil$ denotes
the smallest integer not less than $x \in \mymathbb{R}$.
As usual, $\arccos x \in [0,\pi]$ and $\arcsin x \in [-\pi/2,\pi/2]$ for $x\in [-1,1]$.
The Hermitian conjugate of a matrix $U$ is denoted by $U^{\dagger}$.

Throughout, $I$ denotes the $2\times 2$ identity matrix;
$X,Y$, and $Z$ denote the following Pauli matrices:
\[
X=\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix},
\quad
Y=\begin{pmatrix}
0 & -i} %{{\rm i} \\
i} %{{\rm i} & 0
\end{pmatrix},
\quad
Z=\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix} .
\]
We shall work with a matrix
\begin{equation}
R_{\myv{v} %\underline{v}} %{n}}(\theta) := (\cos\mfrac{\theta}{2}) I - i} %{{\rm i}
(\sin\mfrac{\theta}{2})(v} %\underline{v}} %{n_x X + v} %\underline{v}} %{n_y Y + v} %\underline{v}} %{n_z Z) \label{eq:Rtheta}
\end{equation}
where
$\myv{v} %\underline{v}} %{n}=(v} %\underline{v}} %{n_x,v} %\underline{v}} %{n_y,v} %\underline{v}} %{n_z)^{\rm T} \in S^2$
and $\theta\in\mymathbb{R}$.
This represents the rotation about $\myv{v} %\underline{v}} %{n}$ by angle $ \theta$ (through the homomorphism in Section~\ref{sssub:homomorphism}).
In particular, for $\myv{y}=(0,1,0)^{\rm T}$ and $\myv{z}=(0,0,1)^{\rm T}$, we put
\[
R_{y}(\theta):=R_{\myv{y}}(\theta)
=
\begin{pmatrix}
\cos\frac{\theta}{2} \ & - \sin\frac{\theta}{2} \\
\mbox{}\ \sin\frac{\theta}{2}  &  \cos\frac{\theta}{2}
\end{pmatrix}
\quad \mbox{and}\quad R_{z}(\theta):=R_{\myv{z}}(\theta)
=
\begin{pmatrix}
e} %{{\rm e}^{-i} %{{\rm i}\frac{\theta}{2}} & 0 \\
0 & e} %{{\rm e}^{i} %{{\rm i}\frac{\theta}{2}}
\end{pmatrix} .
\]

For $\myv{m},\myv{\varvn}\in S^2$ with $\abs{\rinnpr{\myv{m}}{\myv{\varvn}}} < 1$, we define the following:
\iffalse \footnote{%
As a definition, $N_{\myv{m},\myv{n}}(U)$ is to be understood as $+\infty$ when the integer set to be minimised is empty,
though the theorem tells this never happens.}
\fi
\begin{equation}
N_{\myv{m},\myv{n}}(U) :=
\min \{  \varnu \in \mymathbb{N} \mid \exists V_1,
\dots,V_{\varnu}\in \mathcal{R}} %{\delta_{\myv{m}} \cup \mathcal{R}} %{\delta_{\myv{\varvn}},\ U= V_1
\cdots V_{\varnu} \} \label{eq:Nmn}
\end{equation}
for $U\in {\rm SU}(2)$, where
$
\mathcal{R}} %{\delta_{\myv{v} %\underline{v}} %{n}} := \{ R_{\myv{v} %\underline{v}} %{n}}(\theta) \mid \theta \in \mymathbb{R} \}
$,
and
\begin{equation}
N_{\myv{m},\myv{n}} := N_{\myv{m},\myv{n}}\big({\rm SU}(2)\big) := \min \{ k \in \mymathbb{N} \mid \forall U \in {\rm SU}(2), \ N_{\myv{m},\myv{n}}(U) \le k \}  .
\end{equation}

Using the homomorphism $F$ from ${\rm SU}(2)$ onto ${\rm SO}(3)$ to be defined in Section~\ref{sssub:homomorphism},
we put
$
\hat{\mathcal{R}} %{\delta}_{\myv{v} %\underline{v}} %{n}} := \big\{ F\big(R_{\myv{v} %\underline{v}} %{n}}(\theta)\big) \mid \theta \in \mymathbb{R} \big\}$.
We extend the definition of $N_{\myv{m},\myv{n}}$ to ${\rm SO}(3)$:
\begin{equation}
N_{\myv{m},\myv{n}}(D) :=
\min \{  \varnu \in \mymathbb{N} \mid \exists \genD_1,
\dots,\genD_{\varnu}\in \hat{\mathcal{R}} %{\delta}_{\myv{m}} \cup \hat{\mathcal{R}} %{\delta}_{\myv{\varvn}},\ D= \genD_1
\cdots \genD_{\varnu} \} \label{eq:NmnSO3}
\end{equation}
for $D\in {\rm SO}(3)$
and
\begin{equation}
N_{\myv{m},\myv{n}}\big({\rm SO}(3)\big) := \min \{ k \in \mymathbb{N} \mid \forall D \in {\rm SO}(3), \ N_{\myv{m},\myv{n}}(D) \le k \}  .
\end{equation}

\subsection{The Maximum of the Minimum Number of Constituent Rotations Over All Target Rotations}

This work's results lead to an elementary self-contained proof of the following known theorem (Appendix~\ref{ss:proofthDecomp}).
\begin{theorem}[{\rm Lowenthal~\cite{Lowenthal71,Lowenthal72}}]
\label{th:Decomp}
For any $\myv{m},\myv{\varvn}\in S^2$ with
$\abs{\rinnpr{\myv{m}}{\myv{\varvn}}} < 1$,
\[
N_{\myv{m},\myv{n}}\big({\rm SO}(3)\big) = N_{\myv{m},\myv{n}}\big({\rm SU}(2)\big) =
\Big\lceil \frac{ \pi }{\arccos \abs{\rinnpr{\myv{m}}{\myv{\varvn}}} } \Big\rceil +1.
\]
\iffalse ORG
\[
N_{\myv{m},\myv{n}} = \Big\lceil \frac{ \pi }{\arccos \abs{\rinnpr{\myv{m}}{\myv{\varvn}}} } \Big\rceil +1.
\]
\fi
\end{theorem}

\subsection{Parameterisations of the Elements in SU(2)\label{sssub:para}}
\mbox{}\
The following lemma presents a well-known parameterisation of ${\rm SU}(2)$ elements.
\begin{lemma}\label{lem:para}
For any element
$U\in{\rm SU}(2)$,
there exist some $\alpha} %{\beta,\gamma} %{\delta \in \mymathbb{R}$,
and $\beta} %{\gamma\in [0,\pi]$ such that
\begin{equation}
U=
\begin{pmatrix}
e} %{{\rm e}^{-i} %{{\rm i} \frac{\gamma} %{\delta+\alpha} %{\beta}{2}}\cos\frac{\beta} %{\gamma}{2} &
\mbox{}\, - e} %{{\rm e}^{i} %{{\rm i} \frac{\gamma} %{\delta-\alpha} %{\beta}{2}} \sin\frac{\beta} %{\gamma}{2}\\
e} %{{\rm e}^{- i} %{{\rm i} \frac{\gamma} %{\delta-\alpha} %{\beta}{2}} \sin\frac{\beta} %{\gamma}{2} &
e} %{{\rm e}^{i} %{{\rm i} \frac{\gamma} %{\delta+\alpha} %{\beta}{2}} \cos\frac{\beta} %{\gamma}{2}
\end{pmatrix} = R_{z}(\alpha} %{\beta) R_{y}(\beta} %{\gamma) R_{z}(\gamma} %{\delta) . \label{eq:uni1}
\end{equation}
\end{lemma}

The parameters $\alpha} %{\beta,\beta} %{\gamma$, and $\gamma} %{\delta$ in this lemma are often called Euler angles
\footnote{The restriction of $\beta} %{\gamma$ to $[0,\pi]$ does not seem common. However,
in a straightforward proof of this lemma, $\beta} %{\gamma\in [0,\pi]$ can be chosen so that
$\cos(\beta} %{\gamma/2)=\abs{a}$ and $\sin(\beta} %{\gamma/2)=\abs{b}$ when the first row of $U$ is $(a,b)$.
Also any $R_{z}(\alpha} %{\beta') R_{y}(\beta} %{\gamma') R_{z}(\gamma} %{\delta')$ without this restriction
can be written as $R_{z}(\alpha} %{\beta) R_{y}(\beta} %{\gamma) R_{z}(\gamma} %{\delta)$ with some $\beta} %{\gamma\in [0,\pi]$ and $\alpha} %{\beta,\gamma} %{\delta\in\mymathbb{R}$.
This readily follows from
equations
$R_{\myv{v}}(\theta+2\pi)=-R_{\myv{v}}(\theta)$, $\myv{v}\in S^2$, $\theta\in\mymathbb{R}$,
and $R_z(-\pi)R_y(\beta} %{\gamma')R_{z}(\pi)=R_y(-\beta} %{\gamma')$, $\beta} %{\gamma'\in\mymathbb{R}$.
\label{fn:pipi}}
The lemma can be rephrased as follows:
Any matrix in ${\rm SU}(2)$ can be
written as
\begin{equation}
\begin{pmatrix}
a & b  \\
- b^* & a^*
\end{pmatrix}\label{eq:ab}
\end{equation}
with some complex numbers $a$ and $b$ such that $|a|^2+|b|^2=1$~\cite{Wigner}.
Hence, any matrix in ${\rm SU}(2)$ can be written as
\begin{equation}
\begin{pmatrix}
w + i} %{{\rm i} z & y + i} %{{\rm i} x \\
- y + i} %{{\rm i} x  &   w - i} %{{\rm i} z
\end{pmatrix}
=
w I + i} %{{\rm i} ( x X + y Y + z Z )
  \label{eq:Uxyzw}
\end{equation}
with some real numbers $x,y,z$, and $w$ such that
$
w^2+x^2+y^2+z^2=1
$.
Take a real number $\theta$ such that $\cos(\theta/2)=w$
and $\sin(\theta/2)=\sqrt{1-w^2}=\sqrt{x^2+y^2+z^2}$; write $x,y$, and $z$
as $x=-v_x\sin(\theta/2),y=-v_y\sin(\theta/2)$, and
$z=-v_z\sin(\theta/2)$, where $v_x,v_y,v_z\in\mymathbb{R}$ and $v_x^2+v_y^2+v_z^2=1$.
Thus, using real numbers $\theta,v_x,v_y,v_z\in\mymathbb{R}$
with $v_x^2+v_y^2+v_z^2=1$,
any matrix in ${\rm SU}(2)$ can be written as
\[
(\cos\mfrac{\theta}{2}) I - i} %{{\rm i} (\sin\mfrac{\theta}{2})
(v_x X + v_y Y + v_z Z) ,
\]
which is nothing but $R_{\myv{v}}(\theta)$ in (\ref{eq:Rtheta}).

\subsection{Homomorphism from SU(2) onto SO(3)\label{sssub:homomorphism}}

For $U \in {\rm SU}(2)$,
we
denote by $F(U)$ the matrix of the linear transformation on $\mymathbb{R}^3$ that sends
$(x,y,z)^{\rm T}$ to $(x',y',z')^{\rm T}$ through
\begin{equation}\label{eq:Uconj}
U(xX+yY+zZ)U^{\dagger}=x'X+y'Y+z'Z .
\end{equation}
Namely, for any $(x,y,z)^{\rm T},(x',y',z')^{\rm T} \in \mymathbb{R}^3$ with (\ref{eq:Uconj}),
\[
\begin{pmatrix}
x' \\ y' \\ z'
\end{pmatrix}
= F(U)
\begin{pmatrix}
x \\ y \\ z
\end{pmatrix} .
\]
We also define
\begin{equation}
\hat{R}_{\myv{v}}(\theta):=
F\big(R_{\myv{v}}(\theta)\big), \quad
\myv{v}\in S^2,\theta\in\mymathbb{R}.
\end{equation}

\subsection{Generic Orthogonal Axes and Coordinate Axes\label{sssub:lmyz}}
Lemma~\ref{lem:para} can be generalised as follows.
\begin{lemma}\label{lem:paraEulerG}
Let $\myv{l},\myv{m}\in S^2$ be vectors with $\rinnprsp{\myv{l}}{\myv{m}}=0$. Then,
for any $V\in {\rm SU}(2)$,
there exist some
$\alpha} %{\beta,\gamma} %{\delta\in\mymathbb{R}$, and $\beta} %{\gamma\in [0,\pi]$ such that
\begin{equation}
V= R_{\myv{m}}(\alpha} %{\beta) R_{\myv{l}}(\beta} %{\gamma) R_{\myv{m}}(\gamma} %{\delta) .
 \label{eq:Eulerlm}
\end{equation}
\end{lemma}

} %{\noindent%
{\em Proof.} \
Since $F$ is onto ${\rm SO}(3)$,
there exists an element $U \in {\rm SU}(2)$ such that
$\myv{l}=F(U) (0,1,0)^{\rm T} \quad\mbox{and}\quad \myv{m}=F(U) (0,0,1)^{\rm T}$.%
\footnote{For the sake of constructiveness,
such an element $U$ is constructed in \refappcomma{app:pr_lm}.}
With this element $U$,
some $\alpha} %{\beta,\gamma} %{\delta\in\mymathbb{R}$, and some $\beta} %{\gamma\in [0,\pi]$,
write
$U^{\dagger} V U = R_{z}(\alpha} %{\beta) R_{y}(\beta} %{\gamma) R_{z}(\gamma} %{\delta)$
in terms of the parameterisation (\ref{eq:uni1}).
Then, since $UR_{z}(\alpha} %{\beta)U^{\dagger}=R_{\myv{m}}(\alpha} %{\beta)$
$UR_{y}(\beta} %{\gamma)U^{\dagger}=R_{\myv{l}}(\beta} %{\gamma)$,
and $UR_{z}(\gamma} %{\delta)U^{\dagger}=R_{\myv{m}}(\gamma} %{\delta)$,
we obtain (\ref{eq:Eulerlm}).
\hfill $\Box$\vspace*{1ex}

We also have Lemma~\ref{lem:transR3}, which is easy but worth recognising.

\begin{lemma}\label{lem:transR3}
Let arbitrary $\kappa, \nu \in \mymathbb{N}$,
$\myv{u}_1,\dots,\myv{u}_\kappa,\myv{v}_1,\dots,\myv{v}_{\nu} \in S^2$,
and $U\in {\rm SU}(2)$ be given.
Put
$\myv{u}_1' = F(U)\myv{u}_1,\dots,\myv{u}_\kappa'=F(U)\myv{u}_\kappa,
\myv{v}_1' = F(U)\myv{v}_1,\dots$, and $\myv{v}_\nu'=F(U)\myv{v}_{\nu}$.
Then, for any
$\theta_1,\dots, \theta_{\kappa},\phi_1,\dots \phi_{\nu} \in \mymathbb{R}$,
\[
R_{\myv{u}_1}(\theta_1) \cdots R_{\myv{u}_\kappa}(\theta_\kappa)
= R_{\myv{v}_1}(\phi_1) \cdots R_{\myv{v}_{\nu}}(\phi_{\nu})
\]
if and only if (iff)
\[
R_{\myv{u}'_1}(\theta_1) \cdots R_{\myv{u}'_\kappa}(\theta_\kappa)
= R_{\myv{v}'_1}(\phi_1) \cdots R_{\myv{v}'_{\nu}}(\phi_{\nu}) .
\]
\end{lemma}

\mynoinden
{\em Proof.}\/ This readily
follows from $UR_{\myv{u}_j}(\theta_j)U^{\dagger}=R_{\myv{u}'_j}(\theta_j)$
and $UR_{\myv{v}_j}(\phi_j)U^{\dagger}=R_{\myv{v}'_j}(\phi_j)$.
\hfill $\Box$} % \vspace*{1ex}

\begin{comment}
This lemma will be used often to rewrite
a statement involved with rotations about generic vectors
as a statement involved with rotations about coordinate axes,
when the latter is easier to prove.
\end{comment}

\begin{comment}
{\em Example.}\/
To illustrate the usefulness of reducing a statement with a generic vector to that with a coordinate axis, we shall check that
$F\big(R_{\myv{m}}(\theta)\big)$ is actually a rotation about $\myv{m}$ by angle $\theta$
in the Euclidean space
for any $\myv{m}\in S^2$ and $\theta\in \mymathbb{R}$.
To see this, putting $\myv{z}=(0,0,1)^{\rm T}$, calculate $F\big(R_{\myv{z}}(\theta)\big)$
as
\begin{equation}
F\big(R_{\myv{z}}(\theta)\big)=F\big(R_z(\theta)\big)
=
\begin{pmatrix}
\cos\theta & -\sin\theta & 0 \\
\sin\theta & \cos\theta & 0 \\
0 & 0 & 1
\end{pmatrix} , \label{eq:rotRz}
\end{equation}
which shows that $F\big(R_{\myv{z}}(\theta)\big)$ is actually the rotation about $\myv{z}$ by angle $\theta$.
Then, since there exists
an element $U\in {\rm SU}(2)$ such that $\myv{m}=F(U) \myv{z}$,
with which $R_{\myv{m}}(\theta)$ can be written as $U R_{\myv{z}}(\theta)U^{\dagger}$,
we have $F\big(R_{\myv{m}}(\theta)\big)=F(UR_{\myv{z}}(\theta)U^{\dagger})= F(U) F\big(R_{\myv{z}}(\theta)\big) [F(U)]^{-1}$.
Regarding the last three rotations, note that $[F(U)]^{-1}$ moves $\myv{m}$ to $\myv{z}$, i.e.,
$[F(U)]^{-1}\myv{m}=\myv{z}$, and $F\big(R_{\myv{z}}(\theta)\big)$ is the rotation about the resultant vector $\myv{z}$ by $\theta$,
and $F(U)$ moves $\myv{z}$ back to $\myv{m}$.
This implies
that $F\big(R_{\myv{m}}(\theta)\big)$ is the desired rotation about $\myv{m}$ by $\theta$.
\end{comment}

\section{%
The Minimum Numbers of Constituent Rotations and
Optimal Constructions of an Arbitrary Rotation
\label{ss:const}}

Here
we present the result establishing $N_{\myv{m},\myv{n}}(U)$ with needed definitions.

\begin{definition}
For $\myv{v}\in S^2$ and
\begin{equation}
U=
\begin{pmatrix}
w + i} %{{\rm i} z & y + i} %{{\rm i} x \\
- y + i} %{{\rm i} x  &   w - i} %{{\rm i} z
\end{pmatrix}
=
wI + i} %{{\rm i} ( x X + y Y + z Z ) \ \ \in \ {\rm SU}(2)
\label{eq:u123}
\end{equation}
where $w,x,y,z\in \mymathbb{R}$ are parameters to
express $U$ uniquely,
$\myb(\myv{v},U)$ is defined by
\begin{equation}
\myb(\myv{v},U) := \abs{(x,y,z)\myv{v}} .
\label{eq:myb}
\end{equation}
\end{definition}

\begin{definition}\label{def:g}
Functions $f: \mymathbb{R}^3 \to [0,\pi]$ and $g: \mymathbb{R}^2
\times (0,\pi/2] \to \mymathbb{N}$ are defined by
\begin{align*}
f& (\alpha} %{\beta,\beta} %{\gamma,\delta):= \\
& 2 \arccos \sqrt{
\cos^2\frac{\beta} %{\gamma}{2}\cos^2\frac{\delta}{2}+
\sin^2\frac{\beta} %{\gamma}{2}\sin^2\frac{\delta}{2}
\mbox{}+2\cos\alpha} %{\beta
\sin\frac{\beta} %{\gamma}{2}\sin\frac{\delta}{2}
\cos\frac{\beta} %{\gamma}{2}\cos\frac{\delta}{2} }
\end{align*}
and
\[
g(\alpha} %{\beta,\beta} %{\gamma,\delta) :=
\begin{cases}
\mbox{}\  {\displaystyle 2\Big\lceil \frac{f(\alpha} %{\beta,\beta} %{\gamma,\delta)}{2\delta} + \frac{1}{2} \Big\rceil }
  & \mbox{if $f(\alpha} %{\beta,\beta} %{\gamma,\delta) \ge \delta$}\\
\mbox{}\ 4 & \mbox{otherwise.}
\end{cases}
\]
\end{definition}

\begin{theorem}\label{th:num_rot}
For any $\myv{m},\myv{\varvn}\in S^2$ with
$\rinnpr{\myv{m}}{\myv{n}} \in [0,1)$,
$\alpha} %{\beta,\gamma} %{\delta\in\mymathbb{R}$, and $\beta} %{\gamma\in [0,\pi]$, if
\[
\myb(\myv{m},U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}) \ge \myb(\myv{n},U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}) ,
\]
then
\[
N_{\myv{m},\myv{n}}\big(F(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})\big) =
N_{\myv{m},\myv{n}}(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})
=
\min \Big\{ 2 \Big\lceil \frac{\beta}{2\delta} \Big\rceil +1 , \,
g(\alpha} %{\beta,\beta} %{\gamma,\delta) , \,
g(\gamma} %{\delta,-\beta} %{\gamma,\delta)
 \Big\}
\]
where
$\delta=\arccos\rinnpr{\myv{m}}{\myv{\varvn}} \in (0,\pi/2]$,
$\myv{l} =\| \myv{m} \times \myv{n} \|^{-1} \myv{m} \times \myv{n}$, and
\[
U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}} := R_{\myv{m}}(\alpha} %{\beta)R_{\myv{l}}(\beta} %{\gamma)
R_{\myv{m}}(\gamma} %{\delta) .
\]
\end{theorem}

Note that there is no loss of generality in assuming
$\myb(\myv{m},U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}) \ge$
$\myb(\myv{n},U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})$,
but also note that $\alpha} %{\beta,\beta} %{\gamma$ and $\gamma} %{\delta$ vary, in general, if $\myv{m}$ and $\myv{n}$ are interchanged.

We give two constructions or decompositions, which will turn out to attain the minimum number
$N_{\myv{m},\myv{n}}(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})$ in the theorem.

\begin{proposition}\label{lem:const_odd}
Given arbitrary
$\myv{m},\myv{\varvn}\in S^2$ with $\rinnpr{\myv{m}}{\myv{n}} \in [0,1)$,
$\alpha} %{\beta,\gamma} %{\delta\in\mymathbb{R}$, and $\beta} %{\gamma\in [0,\pi]$,
put
\begin{equation}\label{eq:premise_delta}
\delta=\arccos\rinnpr{\myv{m}}{\myv{\varvn}}\in (0,\pi/2]
\end{equation}
and
\[
\myv{l} =\| \myv{m} \times \myv{n} \|^{-1} \myv{m} \times \myv{n}.
\]
Then, for any $k\in\mymathbb{N}$ and $\beta} %{\gamma_1, \dots, \beta} %{\gamma_k \in (0, 2\delta]$ satisfying
\begin{equation}\label{eq:beta1k}
\beta} %{\gamma=\beta} %{\gamma_1 + \cdots +\beta} %{\gamma_k ,
\end{equation}
there exist some $\alpha} %{\beta_j,\gamma} %{\delta_j,\theta_j\in\mymathbb{R}$ such that
\begin{equation}\label{eq:pro_RRRR}
R_{\myv{l}}(\beta} %{\gamma_j )=
R_{\myv{m}}(-\alpha} %{\beta_j) R_{\myv{n}}(\theta_j)
 R_{\myv{m}}(-\gamma} %{\delta_j)
\end{equation}
for $j=1,\dots,k$. For these parameters, it holds that
\begin{align}
 R_{\myv{m}}(\alpha} %{\beta)&  R_{\myv{l}}(\beta} %{\gamma) R_{\myv{m}}(\gamma} %{\delta) = \notag\\
& R_{\myv{m}}(\alpha} %{\beta-\alpha} %{\beta_1)R_{\myv{n}}(\theta_1)
 R_{\myv{m}}(-\gamma} %{\delta_1-\alpha} %{\beta_2)R_{\myv{n}}(\theta_2)
 R_{\myv{m}}(-\gamma} %{\delta_2-\alpha} %{\beta_3)R_{\myv{n}}(\theta_3) \cdots \notag\\
& \cdot R_{\myv{m}}(-\gamma} %{\delta_{k-1}-\alpha} %{\beta_{k})R_{\myv{n}}(\theta_{k})
 R_{\myv{m}}(-\gamma} %{\delta_{k}+\gamma} %{\delta) .\label{eq:pro_prod}
\end{align}
\end{proposition}

\begin{remark}\label{rem:const}
The least value of $k$ such that (\ref{eq:beta1k}) holds for some
$\beta} %{\gamma_1, \dots, \beta} %{\gamma_k \in (0,2\delta]$ is $\lceil \beta} %{\gamma/(2\delta) \rceil $.%
\footnote
To make the construction explicit, one can set $\beta} %{\gamma_j = 2\delta$ for $j \ne k$.
The analogous comment applies to the division of $\beta} %{\gamma'+\delta$ in Proposition~\ref{lem:const_even}.}
Hence, this proposition gives a decomposition of an arbitrary element
$U = R_{\myv{m}}(\alpha} %{\beta)  R_{\myv{l}}(\beta} %{\gamma) R_{\myv{m}}(\gamma} %{\delta) \in {\rm SU}(2)$
into the product of
$2\lceil \beta} %{\gamma/(2\delta) \rceil +1$ rotations.%
\footnote{All remarks except Remark~\ref{rem:const}, which needs no proof,
will be proved in what follows.}
\end{remark}

\begin{remark}\label{rem:const_expl}
For $\mtilde{\beta} %{\gamma},\delta\in\mymathbb{R}$ with $0 \le \mtilde{\beta} %{\gamma}/2 \le \delta \le \pi/2$, $\delta \ne 0$,
and $t \in \mymathbb{R}$, let
\[
H_t(\mtilde{\beta} %{\gamma},\delta) :=
\begin{cases}
\mbox{}\ 0 & \mbox{if $\mtilde{\beta} %{\gamma}/2 <\delta = \pi/2$}\\
\mbox{}\ t & \mbox{if $\mtilde{\beta} %{\gamma}/2 = \delta = \pi/2$}\\
{\displaystyle \arcsin \frac{\tan (\mtilde{\beta} %{\gamma}/2) }{\tan \delta}} & \mbox{otherwise.}
\end{cases}
\]
Then, an explicit instance of the set of parameters $\alpha} %{\beta_j, \gamma} %{\delta_j$, and $\theta_j$
for which
(\ref{eq:pro_RRRR})
holds is given by $(\alpha} %{\beta_j,\gamma} %{\delta_j,\theta_j)^{\rm T}=\myvec{\sigma}_{t_j}(\beta} %{\gamma_j,\delta)$,
where
\begin{equation}
\myvec{\sigma}_t(\beta} %{\gamma,\delta):=
\begin{pmatrix}
H_t(\beta} %{\gamma, \delta) - \slfrac{\pi}{2}\\
H_t(\beta} %{\gamma, \delta) + \slfrac{\pi}{2}\\
\mbox{}\ 2 \arcsin \frac{\displaystyle \sin (\beta} %{\gamma/2) }{\displaystyle \sin \delta}
\end{pmatrix}
\end{equation}
and $t_j\in\mymathbb{R}$ can be chosen arbitrarily, $j=1,\dots,k$.
[These make (\ref{eq:pro_prod}) hold.]
\end{remark}

\begin{proposition}\label{lem:const_even}
Given any $\myv{m},\myv{\varvn}\in S^2$ with $\rinnpr{\myv{m}}{\myv{n}} \in [0,1)$,
put $\delta=\arccos\rinnpr{\myv{m}}{\myv{\varvn}}\in (0,\pi/2]$ and
$
\myv{l} =\| \myv{m} \times \myv{n} \|^{-1} \myv{m} \times \myv{n}
$.
For an arbitrary $U \in {\rm SU}(2)$,
choose parameters $\alpha} %{\beta',\gamma} %{\delta'\in \mymathbb{R}$, and $\beta} %{\gamma'\in [0,\pi]$ such that
\begin{equation}\label{eq:const_even}
R_{\myv{l}}(-\delta)U = R_{\myv{m}}(\alpha} %{\beta')  R_{\myv{l}}(\beta} %{\gamma') R_{\myv{m}}(\gamma} %{\delta') .
\end{equation}
Then,
\begin{equation} \label{eq:pro_prod_even0}
U  =  R_{\myv{n}}(\alpha} %{\beta')  R_{\myv{l}}(\beta} %{\gamma'+\delta) R_{\myv{m}}(\gamma} %{\delta')  .
\end{equation}
Furthermore,
for any $k' \in \mymathbb{N}$ and $\beta} %{\gamma'_1, \dots, \beta} %{\gamma'_{k'} \in (0, 2\delta]$ satisfying
\begin{equation}\label{eq:beta1kprime}
\beta} %{\gamma'+\delta=\beta} %{\gamma'_1 + \cdots +\beta} %{\gamma'_{k'} ,
\end{equation}
there exist some $\alpha} %{\beta'_j, \gamma} %{\delta'_j, \theta'_j\in\mymathbb{R}$ such that
\begin{equation}\label{eq:pro_RRRR_even}
R_{\myv{l}}(\beta} %{\gamma'_j )=
R_{\myv{m}}(-\alpha} %{\beta'_j) R_{\myv{n}}(\theta'_j)
 R_{\myv{m}}(-\gamma} %{\delta'_j)
\end{equation}
for $j=1,\dots,k'$. For these parameters, it holds that
\begin{align}
U = & \
R_{\myv{n}}(\alpha} %{\beta') R_{\myv{m}}(-\alpha} %{\beta'_1)R_{\myv{n}}(\theta'_1)
 R_{\myv{m}}(-\gamma} %{\delta'_1-\alpha} %{\beta'_2)R_{\myv{n}}(\theta'_2)
 R_{\myv{m}}(-\gamma} %{\delta'_2-\alpha} %{\beta'_3)R_{\myv{n}}(\theta'_3) \cdots \notag\\
& \
 \cdot R_{\myv{m}}(-\gamma} %{\delta'_{k'-1}-\alpha} %{\beta'_{k'})R_{\myv{n}}(\theta'_{k'})
 R_{\myv{m}}(-\gamma} %{\delta'_{k'}+\gamma} %{\delta') . \label{eq:pro_prod_even}
\end{align}
\end{proposition}

\begin{remark}\label{rem:const_even}
The least value of $k'$ such that (\ref{eq:beta1kprime}) holds for some
$\beta} %{\gamma'_1, \dots, \beta} %{\gamma'_{k'} \in (0,2\delta]$ is
$\lceil  (\beta} %{\gamma' +\delta)/ (2\delta) \rceil
=\lceil  \beta} %{\gamma' / (2\delta) + 1/2 \rceil$.
Moreover, if $\beta} %{\gamma' \ge \delta$ and $k'=\lceil \beta} %{\gamma'/(2\delta) + 1/2 \rceil $,
the parameter $\alpha} %{\beta'_1$
can be chosen
so that it satisfies $\alpha} %{\beta'_1=0$
as well as
(\ref{eq:pro_RRRR_even}) and (\ref{eq:pro_prod_even}).
Hence, when $\beta} %{\gamma' \ge \delta$, this proposition and the fact just mentioned give
a decomposition of an arbitrary element
$U = R_{\myv{n}}(\alpha} %{\beta')  R_{\myv{l}}(\beta} %{\gamma'+\delta) R_{\myv{m}}(\gamma} %{\delta') \in {\rm SU}(2)$
into the product of
$2 \lceil  \beta} %{\gamma' / (2\delta) + 1/2 \rceil$ rotations,
and when $\beta} %{\gamma' < \delta$, a decomposition of
$U$ into the product of
four rotations.
\end{remark}

\begin{remark}\label{rem:const_expl_even}
An explicit instance of the set of parameters $\alpha} %{\beta'_j, \gamma} %{\delta'_j$, and $\theta'_j$, $j=1,\dots,k'$, for which
(\ref{eq:pro_RRRR_even}) and (\ref{eq:pro_prod_even}) hold is given by
$(\alpha} %{\beta'_j,\gamma} %{\delta'_j,\theta'_j)^{\rm T}=\myvec{\sigma}_{t_j}(\beta} %{\gamma'_j,\delta)$,
where $t_j\in\mymathbb{R}$ can be chosen arbitrarily, $j=1,\dots,k'$.
\end{remark}

\section{%
Limits on Constructions \label{ss:limits}}

In order to bound $N_{\myv{m},\myv{n}}( D )$, etc., from below,
we use the geodesic metric on the unit sphere $S^2$,
which is denoted by $d$.
Specifically,
\begin{equation}
d(\myv{u},\myv{v}) := \arccos \rinnpr{\myv{u}}{\myv{v}}
\in [0,\pi]
\end{equation}
for
$\myv{u},\myv{v}\in S^2$. This is the length of
the geodesic connecting $\myv{u}$ and $\myv{v}$ on $S^2$.
We have the following lemma.
[Recall we have put $\hat{R}_{\myv{v}}(\theta)=
F\big(R_{\myv{v}}(\theta)\big)$.]

\begin{lemma}\label{lem:ti}
Let $\myv{n},\myv{m}$ be arbitrary
vectors in $S^2$ with
$\delta = d(\myv{m},\myv{n}) = \arccos \rinnpr{\myv{m}}{\myv{n}} \in (0,\pi]$.
Then,
for any $k} %{j \in \mymathbb{N}$ and $\phi} %{\sigma_1,\dots,\phi} %{\sigma_{2k} %{j}\in\mymathbb{R}$, the following inequalities hold:
\begin{gather}
d(\hat{R}_{\myv{m}}(\phi} %{\sigma_{2k} %{j-1}) \hat{R}_{\myv{n}}(\phi} %{\sigma_{2k} %{j-2}) \cdots \hat{R}_{\myv{m}}(\phi} %{\sigma_3)
\hat{R}_{\myv{n}}(\phi} %{\sigma_{2})\hat{R}_{\myv{m}}(\phi} %{\sigma_1)\myv{m}, \myv{m}) \le 2 (k} %{j-1) \delta,  \label{eq:lem3_3}\\
d(\hat{R}_{\myv{m}}(\phi} %{\sigma_{2k} %{j-1}) \hat{R}_{\myv{n}}(\phi} %{\sigma_{2k} %{j-2}) \cdots \hat{R}_{\myv{m}}(\phi} %{\sigma_3)
\hat{R}_{\myv{n}}(\phi} %{\sigma_{2}) \hat{R}_{\myv{m}}(\phi} %{\sigma_1)\myv{m}, \myv{n}) \le (2 k} %{j -1) \delta, \label{eq:lem3_4}\\
d(\hat{R}_{\myv{n}}(\phi} %{\sigma_{2k} %{j}) \hat{R}_{\myv{m}}(\phi} %{\sigma_{2k} %{j-1}) \cdots \hat{R}_{\myv{m}}(\phi} %{\sigma_3)
\hat{R}_{\myv{n}}(\phi} %{\sigma_{2}) \hat{R}_{\myv{m}}(\phi} %{\sigma_1)\myv{m}, \myv{n}) \le (2 k} %{j -1) \delta , \label{eq:lem3_2} \\
d(\hat{R}_{\myv{n}}(\phi} %{\sigma_{2k} %{j}) \hat{R}_{\myv{m}}(\phi} %{\sigma_{2k} %{j-1}) \cdots \hat{R}_{\myv{m}}(\phi} %{\sigma_3)
\hat{R}_{\myv{n}}(\phi} %{\sigma_{2}) \hat{R}_{\myv{m}}(\phi} %{\sigma_1)\myv{m}, \myv{m}) \le 2 k} %{j \delta. \label{eq:lem3_1}
\end{gather}
\end{lemma}
This can be shown
easily by induction on $k} %{j$
using the triangle inequality for $d$.
In what follows,
(\ref{eq:lem3_3}) and (\ref{eq:lem3_2}) will be used in the following forms:
\begin{equation}
2\Big\lceil \frac{d(D\myv{m},\myv{m})}{2\delta} \Big\rceil +1 \le 2 k} %{j -1 \quad\mbox{and}\quad
2\Big\lceil \frac{d(D'\myv{m},\myv{n})}{2\delta} +\frac{1}{2}\Big\rceil  \le 2 k} %{j .
\end{equation}
These bounds hold when $D$ and $D' \in {\rm SO}(3)$ equal the product of $2k} %{j-1$ rotations and that of $2k} %{j$ rotations, respectively, in Lemma~\ref{lem:ti} (since $k} %{j$ is an integer).
It will turn out that these bounds are tight.

\section{Proof of the Results \label{ss:Proofs}}

\subsection{Structure of the Proof}

Here the structure of the whole proof of the results in this work is described.
Theorem~\ref{th:num_rot} is obtained as a consequence of Lemma~\ref{lem:1} to be presented.
The constructive half of Lemma~\ref{lem:1} is due to Propositions~\ref{lem:const_odd} and \ref{lem:const_even}.
The other half of Lemma~\ref{lem:1}, related to limits on constructions,
is due to Lemma~\ref{lem:ti}.
Theorem~\ref{th:Decomp} is derived from Theorem~\ref{th:num_rot} in Appendix~\ref{ss:proofthDecomp}.

\subsection{Proof of Propositions~\ref{lem:const_odd} and \ref{lem:const_even}}

The following lemma is fundamental to the results in this work.

\begin{lemma}\label{lem:EulerG}
For any $\beta} %{\gamma,\theta\in\mymathbb{R}$ and
for any $\myv{\varu}, \myv{l},\myv{m} \in S^2$
such that $\rinnprsp{\myv{l}}{\myv{m}}=0$,
the following two conditions are equivalent.\vspace{1ex}

\mbox{}\hspace{.5ex}I. There exist some $\alpha} %{\beta,\gamma} %{\delta\in\mymathbb{R}$ such that
\begin{equation}\label{eq:EulerG}
R_{\myv{\varu}}(\theta)
= R_{\myv{m}}(\alpha} %{\beta) R_{\myv{l}}(\beta} %{\gamma)
 R_{\myv{m}}(\gamma} %{\delta) .
\end{equation}

\mbox{}\hspace{.5ex}II. $\sqrt{1-(\rinnpr{\myv{m}}{\myv{\varu}})^2}\abs{\sin\sfrac{\theta}{2}}
 =\abs{\sin\sfrac{\beta} %{\gamma}{2}}$.
\end{lemma}

\noindent
{\em Proof.}

1)
Take an element $U\in {\rm SU}(2)$ such that
\begin{equation}
\myv{l}=F(U) (0,1,0)^{\rm T} \quad\mbox{and}\quad \myv{m}=F(U) (0,0,1)^{\rm T} ,
\end{equation}
and put $\myv{v}=(v_x,v_y,v_z)^{\rm T}$ for the parameters $v_x,v_y$, and $v_z$ such that
\begin{equation}
\myv{\varu}= v_x \myv{l} \times \myv{m} + v_y \myv{l} + v_z \myv{m}.\label{eq:u_v}
\end{equation}
Then, owing to Lemma~\ref{lem:transR3}, (\ref{eq:EulerG})
holds iff
\begin{equation}
R_{\myv{v}}(\theta) = R_{z}(\alpha} %{\beta) R_{y}(\beta} %{\gamma) R_{z}(\gamma} %{\delta) .
\label{eq:v_zyz}
\end{equation}

2)
A direct calculation shows
\begin{eqnarray}
R_{z}(\alpha} %{\beta) R_{y}(\beta} %{\gamma) R_{z}(\gamma} %{\delta)
&=&    \cos\frac{\beta} %{\gamma}{2} \cos\frac{\gamma} %{\delta+\alpha} %{\beta}{2} I
-i} %{{\rm i}\sin\frac{\beta} %{\gamma}{2} \sin\frac{\gamma} %{\delta-\alpha} %{\beta}{2} X \notag\\
&&  \mbox{}-i} %{{\rm i}\sin\frac{\beta} %{\gamma}{2} \cos\frac{\gamma} %{\delta-\alpha} %{\beta}{2} Y
-i} %{{\rm i}\cos\frac{\beta} %{\gamma}{2} \sin\frac{\gamma} %{\delta+\alpha} %{\beta}{2} Z .\label{eq:zyz}
\end{eqnarray}
Hence, (\ref{eq:v_zyz}) is equivalent to
\begin{numcases}
{}
\cos\mfrac{\theta}{2}  =  \cos\mfrac{\beta} %{\gamma}{2}
  \cos\mfrac{\gamma} %{\delta+\alpha} %{\beta}{2}\label{eq:1}\\
v} %\underline{v}} %{n_x \sin\mfrac{\theta}{2}  =  \sin\mfrac{\beta} %{\gamma}{2}
  \sin\mfrac{\gamma} %{\delta-\alpha} %{\beta}{2} \label{eq:2}\\
v} %\underline{v}} %{n_y \sin\mfrac{\theta}{2}  =  \sin\mfrac{\beta} %{\gamma}{2}
  \cos\mfrac{\gamma} %{\delta-\alpha} %{\beta}{2} \label{eq:3}\\
v} %\underline{v}} %{n_z \sin\mfrac{\theta}{2}  =  \cos\mfrac{\beta} %{\gamma}{2}
  \sin\mfrac{\gamma} %{\delta + \alpha} %{\beta}{2} . \label{eq:4}
\end{numcases}

3) We shall prove I $\Rightarrow$ II.
On each side of (\ref{eq:2}) and (\ref{eq:3}),
squaring and
summing the resultant pair, we have
\begin{equation}
\sqrt{1-v} %\underline{v}} %{n_z^2}\abs{\sin\mfrac{\theta}{2}}=\abs{\sin\mfrac{\beta} %{\gamma}{2}}.\label{eq:IIorg}
\end{equation}
[Eqs.\ (\ref{eq:1}) and (\ref{eq:4}) also imply (\ref{eq:IIorg})
similarly.]
But (\ref{eq:IIorg}) implies II in view of (\ref{eq:u_v}).

4) Next, we shall prove II $\Rightarrow$ I.

Transforming $(\alpha} %{\beta,\beta} %{\gamma)$ into $(\eta,\zeta)$, where the two pairs
are related by
\begin{equation}
\eta=\frac{\gamma} %{\delta+\alpha} %{\beta}{2} \quad\mbox{and}\quad \zeta=\frac{\gamma} %{\delta-\alpha} %{\beta}{2},
 \label{eq:5_6}
\end{equation}
we see, from the paragraphs 1) and 2), that
I is equivalent to the following condition:
There exist some $\eta,\zeta\in\mymathbb{R}$ such that
\begin{numcases}
{}
\cos\mfrac{\theta}{2}  =  \cos\mfrac{\beta} %{\gamma}{2} \cos\eta
\label{eq:7} \\
v} %\underline{v}} %{n_x \sin\mfrac{\theta}{2}  =  \sin\mfrac{\beta} %{\gamma}{2} \sin\zeta
 \label{eq:8}\\
v} %\underline{v}} %{n_y \sin\mfrac{\theta}{2}   =  \sin\mfrac{\beta} %{\gamma}{2}
  \cos\zeta \label{eq:9}\\
v} %\underline{v}} %{n_z \sin\mfrac{\theta}{2}   =  \cos\mfrac{\beta} %{\gamma}{2}
  \sin\eta. \label{eq:10}
\end{numcases}
Hence,
it is enough to
show that
II implies
the existence of some $\eta,\zeta\in\mymathbb{R}$ satisfying (\ref{eq:7})--(\ref{eq:10}).

Now suppose $\cos\mfrac{\beta} %{\gamma}{2} \ne 0$.
Then, if we show
\begin{equation}
\frac{ \cos^2\mfrac{\theta}{2} }{ \cos^2\mfrac{\beta} %{\gamma}{2} } +
\frac{ v} %\underline{v}} %{n_z^2 \sin^2\mfrac{\theta}{2} }{ \cos^2\mfrac{\beta} %{\gamma}{2} } = 1 ,
 \label{eq:pr1}
\end{equation}
it will immediately
imply
the existence of $\eta$ satisfying (\ref{eq:7}) and (\ref{eq:10}).
\begin{comment}
Namely, $\eta$ will be specified by
\begin{equation}
\cos\eta = \frac{ \cos\mfrac{\theta}{2} }{
  \cos\mfrac{\beta} %{\gamma}{2} } \quad\mbox{and}\quad
\sin\eta = \frac{ v} %\underline{v}} %{n_z \sin\mfrac{\theta}{2} }{
  \cos\mfrac{\beta} %{\gamma}{2} }. \label{eq:eta_tri}
\end{equation}
\end{comment}
From
II, however, we have (\ref{eq:IIorg}), and hence,
$(1-v} %\underline{v}} %{n_z^2)\sin^2\mfrac{\theta}{2}=\sin^2\mfrac{\beta} %{\gamma}{2}$,
i.e.,
$1-(1-v} %\underline{v}} %{n_z^2)\sin^2\mfrac{\theta}{2}=\cos^2\mfrac{\beta} %{\gamma}{2}$,
which is equivalent to (\ref{eq:pr1}) by the assumption
$\cos\mfrac{\beta} %{\gamma}{2} \ne 0$.
If $\cos\mfrac{\beta} %{\gamma}{2} = 0$, then $\abs{\sin\mfrac{\beta} %{\gamma}{2}}=1$.
This and (\ref{eq:IIorg}) imply
$1-v} %\underline{v}} %{n_z^2=\abs{\sin\mfrac{\theta}{2}}=1$, and hence,
$v} %\underline{v}} %{n_z=\cos\mfrac{\theta}{2}=0$.
Then,
(\ref{eq:7}) and (\ref{eq:10}) hold for any choice of $\eta$.

In a similar way, if $\sin\mfrac{\beta} %{\gamma}{2} \ne 0$,
\begin{equation}
\frac{ v} %\underline{v}} %{n_x^2 \sin^2\mfrac{\theta}{2} }{ \sin^2\mfrac{\beta} %{\gamma}{2} } +
\frac{ v} %\underline{v}} %{n_y^2 \sin^2\mfrac{\theta}{2} }{ \sin^2\mfrac{\beta} %{\gamma}{2} } = 1
 \label{eq:pr2}
\end{equation}
will immediately imply
the existence of $\zeta$ satisfying (\ref{eq:8}) and (\ref{eq:9}).
\begin{comment}
Namely, $\zeta$ will be specified by
\begin{equation}
 \sin\zeta = \frac{v} %\underline{v}} %{n_x \sin\mfrac{\theta}{2}}{
  \sin\mfrac{\beta} %{\gamma}{2} } \quad\mbox{and}\quad
\cos\zeta = \frac{v} %\underline{v}} %{n_y \sin\mfrac{\theta}{2}}{
 \sin\mfrac{\beta} %{\gamma}{2} } . \label{eq:pr3}
\end{equation}
\end{comment}
But (\ref{eq:pr2}) follows again from II or (\ref{eq:IIorg})
since $1-v} %\underline{v}} %{n_z^2=v} %\underline{v}} %{n_x^2+v} %\underline{v}} %{n_y^2$.
If $\sin\mfrac{\beta} %{\gamma}{2} = 0$,
both (\ref{eq:8}) and (\ref{eq:9}) hold for any choice of $\zeta$ similarly.
\hfill $\Box$\vspace*{1ex}

} %{\noindent%
{\em Proof of Proposition~\ref{lem:const_odd}.}\/ \hspace*{2.8ex
Choose a parameter
$\theta_j$
such that $\abs{\sin(\theta_j/2)}=$ $\sin(\beta} %{\gamma_j/2)/\sin\delta$,
which is possible
by the assumption $\beta} %{\gamma_j \in (0, 2\delta]$; then,
it follows from Lemma~\ref{lem:EulerG}
that there exist some $\alpha} %{\beta_j,\gamma} %{\delta_j\in \mymathbb{R}$ such that
(\ref{eq:pro_RRRR}), i.e.,
$R_{\myv{l}}(\beta} %{\gamma_j )=
R_{\myv{m}}(-\alpha} %{\beta_j) R_{\myv{n}}(\theta_j)
 R_{\myv{m}}(-\gamma} %{\delta_j)$ holds,
$j=1,\dots,k$.
Inserting these into
\[
R_{\myv{m}}(\alpha} %{\beta)R_{\myv{l}}(\beta} %{\gamma)R_{\myv{m}}(\gamma} %{\delta)=R_{\myv{m}}(\alpha} %{\beta)R_{\myv{l}}(\beta} %{\gamma_1) \cdots R_{\myv{l}}(\beta} %{\gamma_k) R_{\myv{m}}(\gamma} %{\delta),
\]
we obtain (\ref{eq:pro_prod}).
\hfill $\Box$\vspace*{1ex}

} %{\noindent%
{\em Proof of Proposition~\ref{lem:const_even}}.\/
Note $R_{\myv{l}}(\delta)R_{\myv{m}}(\alpha} %{\beta')R_{\myv{l}}(-\delta)= R_{\myv{n}}(\alpha} %{\beta')$, which is equivalent to
$R_{y}(\delta)R_{z}(\alpha} %{\beta')R_{y}(-\delta)=R_{v}(\alpha} %{\beta')$, where $\myv{v} = (\sin\delta, 0, \cos\delta)^{\rm T}$,
by Lemma~\ref{lem:transR3}
Figure~\ref{fig:1}) and therefore, can be checked easily by a direct calculation.
Using this equation,
we can rewrite (\ref{eq:const_even}) as
$U=R_{\myv{n}}(\alpha} %{\beta')  R_{\myv{l}}(\beta} %{\gamma'+\delta) R_{\myv{m}}(\gamma} %{\delta')$, which is (\ref{eq:pro_prod_even0}).
Then, applying to $R_{\myv{l}}(\beta} %{\gamma'+\delta) R_{\myv{m}}(\gamma} %{\delta')$ the decomposition in Proposition~\ref{lem:const_odd} with $(\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta)$ replaced by $(0,\beta} %{\gamma'+\delta,\gamma} %{\delta')$,
it readily follows that there exist some $\alpha} %{\beta'_j,\gamma} %{\delta'_j$, and $\theta'_j \in \mymathbb{R}$,
$j=1,\dots,k'$, that satisfy the following:
$\abs{\sin(\theta'_j/2)}=\sin(\beta} %{\gamma'_j/2)/\sin\delta$ and (\ref{eq:pro_RRRR_even}) for
$j=1,\dots,k'$, and
\begin{align}
R_{\myv{l}}(\beta} %{\gamma'+\delta)& R_{\myv{m}}(\gamma} %{\delta') \notag\\
=& \
 R_{\myv{m}}(-\alpha} %{\beta'_1)R_{\myv{n}}(\theta'_1)
 R_{\myv{m}}(-\gamma} %{\delta'_1-\alpha} %{\beta'_2)R_{\myv{n}}(\theta'_2)
 R_{\myv{m}}(-\gamma} %{\delta'_2-\alpha} %{\beta'_3)R_{\myv{n}}(\theta'_3) \cdots \notag\\
& \
 \cdot R_{\myv{m}}(-\gamma} %{\delta'_{k'-1}-\alpha} %{\beta'_{k'})R_{\myv{n}}(\theta'_{k'})
 R_{\myv{m}}(-\gamma} %{\delta'_{k'}+\gamma} %{\delta') .
\end{align}
\iffalse
\begin{eqnarray}
R_{\myv{l}}(\beta} %{\gamma'+\delta) R_{\myv{m}}(\gamma} %{\delta')
&=&
 R_{\myv{m}}(-\alpha} %{\beta'_1)R_{\myv{n}}(\theta'_1)
 R_{\myv{m}}(-\gamma} %{\delta'_1-\alpha} %{\beta'_2)R_{\myv{n}}(\theta'_2)
 R_{\myv{m}}(-\gamma} %{\delta'_2-\alpha} %{\beta'_3)R_{\myv{n}}(\theta'_3) \cdots \notag\\
&&
 \cdot R_{\myv{m}}(-\gamma} %{\delta'_{k'-1}-\alpha} %{\beta'_{k'})R_{\myv{n}}(\theta'_{k'})
 R_{\myv{m}}(-\gamma} %{\delta'_{k'}+\gamma} %{\delta') .
\end{eqnarray}
\fi
Thus, we obtain the proposition.
\hfill $\Box$\vspace*{1ex}

Remarks~\ref{rem:const_expl} and \ref{rem:const_expl_even} to these propositions
are proved in \refapp{app:proof_rem}.
The statement on $\alpha} %{\beta'_1$ in Remark~\ref{rem:const_even} follows from
Remark~\ref{rem:const_expl_even} (put $\beta} %{\gamma'_1=2\delta$ and $t_1=\pi/2$) or, more directly, from
an equation
$R_{\myv{l}}(2\delta)=R_{\myv{n}}(\pi)R_{\myv{m}}(-\pi)$, which is equivalent to
$R_{y}(2\delta)=R_{v}(\pi)R_{z}(-\pi)$, where $\myv{v}=(\sin\delta,0,\cos\delta)^{\rm T}$,
by Lemma~\ref{lem:transR3}.
\begin{figure}
\begin{center}
\begin{picture}(150,165)(-50,-55)
\put(111,0){\makebox(0,0)[l]{$\myv{z}$\ ($\myv{m}$)}}
\put(0,0){\vector(1,0){109.5}}
\put(0,100){\makebox(0,0)[b]{$\myv{x}$\ ($\myv{l} \times \myv{m}$)}}
\put(0,0){\vector(0,1){98.5}}
\put(1,-4){\makebox(0,0)[t]{$0$}}
\put(0,0){\vector(-1,-1){40}}
\put(-40,-40){\makebox(0,0)[t]{$\myv{y}$\ ($\myv{l}$)}}
\put(0,0){\vector(2,1){98}}
\put(101,50){\makebox(0,0)[l]{$\myv{v}$\ ($\myv{n}$)}}
\multiput(98,0)(0,2){25}{\line(0,1){1}}
\put(92,-2){\makebox(0,0)[t]{$\cos\delta$}}
\multiput(0,49)(2,0){49}{\line(1,0){1}}
\put(-3,50){\makebox(0,0)[r]{$\sin\delta$}}
\end{picture}
\caption{Configuration of $\myv{l},\myv{m}$, and $\myv{n}$ in Propositions~\ref{lem:const_odd} and \ref{lem:const_even},
and configuration of $\myv{y}=(0,1,0)^{\rm T}$, $\myv{z}=(0,0,1)^{\rm T}$, and $\myv{v}$ in arguments around these propositions \label{fig:1}}
\end{center}
\end{figure}

\subsection{Proof of Theorem~\ref{th:num_rot} \label{sssub:ProofTh2}}

Let $2\mymathbb{N} -1$ and $2\mymathbb{N}$ denote the set of odd numbers in $\mymathbb{N}$ and
that of even numbers in $\mymathbb{N}$, respectively.
We define the following for $\myv{m},\myv{\varvn}\in S^2$ with $\abs{\rinnpr{\myv{m}}{\myv{\varvn}}} < 1$:
\begin{align*}
\varM{\myv{m},\myv{n}}{{\rm odd}}(U) :=
\min \{  \varnu \in 2\mymathbb{N} \! - \! 1 \mid \, &
\exists V_1, V_3, \dots,V_{\varnu}\in \mathcal{R}} %{\delta_{\myv{m}},\\
& \exists V_2, V_4, \dots,V_{\varnu-1}\in \mathcal{R}} %{\delta_{\myv{n}}, \
 U=  V_{\varnu}V_{\varnu-1} \cdots V_1\} ,\\
\varM{\myv{m},\myv{n}}{\rm even}(U) :=
\min \{   \varnu \in 2\mymathbb{N} \mid \, &
\exists V_1, V_3, \dots,V_{\varnu-1}\in \mathcal{R}} %{\delta_{\myv{m}},\\
& \exists V_2, V_4, \dots,V_{\varnu}\in \mathcal{R}} %{\delta_{\myv{n}}, \
 U=  V_{\varnu}V_{\varnu-1} \cdots V_1\} ,\\
\varMcomb{\myv{m},\myv{n}}(U) :=
\min \{  \varM{\myv{m},\myv{n}}{{\rm odd}}(U), & \ \varM{\myv{m},\myv{n}}{\rm even}(U) \}
\end{align*}
for $U\in {\rm SU}(2)$;
\begin{align*}
\mbox{}\
\varM{\myv{m},\myv{n}}{\rm odd}(D) :=
\min \{  \varnu \in 2\mymathbb{N} \! - \! 1 \mid \, &
\exists \genD_1, \genD_3, \dots,\genD_{\varnu}\in \hat{\mathcal{R}} %{\delta}_{\myv{m}},\\
& \mbox{}\!\! \exists \genD_2, \genD_4, \dots,\genD_{\varnu-1}\in \hat{\mathcal{R}} %{\delta}_{\myv{n}}, \
D=  \genD_{\varnu}\genD_{\varnu-1} \cdots \genD_1\} ,\\
\varM{\myv{m},\myv{n}}{\rm even}(D) :=
\min \{  \varnu \in 2\mymathbb{N} \mid \, &
\exists \genD_1, \genD_3, \dots,\genD_{\varnu-1}\in \hat{\mathcal{R}} %{\delta}_{\myv{m}},\\
& \exists \genD_2, \genD_4, \dots,\genD_{\varnu}\in \hat{\mathcal{R}} %{\delta}_{\myv{n}}, \
 D=  \genD_{\varnu}\genD_{\varnu-1} \cdots \genD_1\} ,\\
\varMcomb{\myv{m},\myv{n}}(D) :=
\min \{ \varM{\myv{m},\myv{n}}{\rm odd}(D), & \ \varM{\myv{m},\myv{n}}{\rm even}(D) \}
\end{align*}
for $D\in {\rm SO}(3)$.
The following lemma largely solves the issue of determining
the optimal number $N_{\myv{m},\myv{n}}(U)$.

\begin{lemma}\label{lem:1}
Let $\myv{m},\myv{n}$, $\myv{l}$, and $\delta$
be as in Theorem~\ref{th:num_rot}.
Then, for any $\alpha} %{\beta,\gamma} %{\delta\in\mymathbb{R}$, and $\beta} %{\gamma\in [0,\pi]$,
\begin{equation}
\varM{\myv{m},\myv{n}}{\rm odd}\big(F(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})\big) =
\varM{\myv{m},\myv{n}}{\rm odd}(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}) =
2 \Big\lceil \frac{\beta}{2\delta} \Big\rceil +1 \label{eq:lem1_1}
\end{equation}
and
\begin{equation}
\varM{\myv{m},\myv{n}}{\rm even}\big(F(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})\big) =
\varM{\myv{m},\myv{n}}{\rm even}(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}) =
g(\alpha} %{\beta,\beta} %{\gamma,\delta) \label{eq:lem1_2}
\end{equation}
where
$U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}$
is as defined in Theorem~\ref{th:num_rot}.
\end{lemma}
\begin{corollary}\label{coro:1}
Let $\myv{m},\myv{n}$, $\myv{l}$, and $\delta$
be as in Theorem~\ref{th:num_rot}.
Then, for any $\alpha} %{\beta,\gamma} %{\delta\in\mymathbb{R}$, and $\beta} %{\gamma\in [0,\pi]$,
\begin{gather}\label{eq:2:DUP_SEP10}
\varMcomb{\myv{m},\myv{n}}\big(F(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})\big)
=
\varMcomb{\myv{m},\myv{n}}(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})
= \min \Big\{ 2 \Big\lceil \frac{\beta}{2\delta} \Big\rceil +1 , \,
g(\alpha} %{\beta,\beta} %{\gamma,\delta) \Big\}.
\end{gather}
\end{corollary}

} %{\noindent%
{\em Proof}.\/
In the case where $\beta} %{\gamma=0$,
since $\varM{\myv{m},\myv{n}}{\rm odd}(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}) = 1$
and
$\varM{\myv{m},\myv{n}}{\rm even}(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}) = 2$,
(\ref{eq:lem1_1}) and (\ref{eq:lem1_2}) are trivially true.
We shall prove the statement for $\beta} %{\gamma>0$.

To establish (\ref{eq:lem1_1}), we shall show the first and third inequalities in
\begin{equation}
2 \Big\lceil \frac{\beta}{2\delta} \Big\rceil +1
\le
\varM{\myv{m},\myv{n}}{\rm odd}\big(F(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})\big)
\le
\varM{\myv{m},\myv{n}}{\rm odd}(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})
\le
2 \Big\lceil \frac{\beta}{2\delta} \Big\rceil +1
\label{eq:lem1_1_comb}
\end{equation}
while the second inequality
trivially follows from the definition of $\varM{\myv{m},\myv{n}}{\rm odd}$.

Note first that Remark~\ref{rem:const} to Proposition~\ref{lem:const_odd} immediately implies
the third inequality in (\ref{eq:lem1_1_comb}).
To prove the first
inequality,
assume
\begin{equation}
F(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})= \genD_{\varnu}\genD_{\varnu-1}\cdots \genD_1
\label{eq:pr_lem1_1}
\end{equation}
for some $j=2k-1$ with $k\in\mymathbb{N}$,
where $\genD_{\nu}\in\hat{\mathcal{R}} %{\delta}_{\myv{m}}$ if $\nu$ is odd
and $\genD_{\nu}\in\hat{\mathcal{R}} %{\delta}_{\myv{n}}$ otherwise.

We shall evaluate $d(F(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}} ) \myv{m},\myv{m})=
d(\genD_{\varnu}\genD_{\varnu-1}\cdots \genD_1 \myv{m},\myv{m})$. \hspace*{1ex} Noting that
$d(F(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})\myv{m},\myv{m})=\beta} %{\gamma$, we have
$\beta} %{\gamma \le 2(k-1)\delta$ by (\ref{eq:lem3_3}) of Lemma~\ref{lem:ti}.
This implies $\lceil \beta} %{\gamma/(2\delta) \rceil \le k-1$, and therefore,
\begin{equation}\label{eq:oddlb}
2\Big\lceil \frac{\beta} %{\gamma}{2\delta} \Big\rceil +1 \le 2k-1=j.
\end{equation}
From this bound,
we have the first inequality in
(\ref{eq:lem1_1_comb}),
and hence, (\ref{eq:lem1_1}).

To establish (\ref{eq:lem1_2}), we shall first treat the major case where $f(\alpha} %{\beta,\beta} %{\gamma,\delta) \ge \delta$.
Recalling that $g(\alpha} %{\beta,\beta} %{\gamma,\delta) = 2 \lceil f(\alpha} %{\beta,\beta} %{\gamma,\delta)/(2\delta) +1/2 \rceil$ in this case, we shall show the first and third inequalities in
\begin{equation}\label{eq:lem1_2_comb}
2 \Big\lceil \frac{f(\alpha} %{\beta,\beta} %{\gamma,\delta)}{2\delta} + \frac{1}{2} \Big\rceil
\le
\varM{\myv{m},\myv{n}}{\rm even}\big(F(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})\big)
\le
\varM{\myv{m},\myv{n}}{\rm even}(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}) \le
2 \Big\lceil \frac{f(\alpha} %{\beta,\beta} %{\gamma,\delta)}{2\delta} + \frac{1}{2} \Big\rceil
\end{equation}
while the second inequality holds trivially.

Note that Remark~\ref{rem:const_even} to Proposition~\ref{lem:const_even} will imply
the third inequality
upon showing that $\beta} %{\gamma'$ in Proposition~\ref{lem:const_even} satisfies $\beta} %{\gamma'=f(\alpha} %{\beta,\beta} %{\gamma,\delta)$
when $U=U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}$.
To see $\beta} %{\gamma'=f(\alpha} %{\beta,\beta} %{\gamma,\delta)$, rewrite (\ref{eq:const_even}), using Lemma~\ref{lem:transR3}, as
\begin{equation}
R_{y}(-\delta)R_{z}(\alpha} %{\beta)R_{y}(\beta} %{\gamma)R_{z}(\gamma} %{\delta) = R_{z}(\alpha} %{\beta')  R_{y}(\beta} %{\gamma') R_{z}(\gamma} %{\delta') .
\end{equation}
Then, a direct calculation shows the absolute value of the $(1,1)$-entry of the left-hand side equals
\[
\sqrt{
\cos^2\frac{\beta} %{\gamma}{2}\cos^2\frac{\delta}{2}+
\sin^2\frac{\beta} %{\gamma}{2}\sin^2\frac{\delta}{2}
\mbox{}+2\cos\alpha} %{\beta
\sin\frac{\beta} %{\gamma}{2}\sin\frac{\delta}{2}
\cos\frac{\beta} %{\gamma}{2}\cos\frac{\delta}{2} }.
\]
This shows $\beta} %{\gamma'=f(\alpha} %{\beta,\beta} %{\gamma,\delta)$ in view of (\ref{eq:uni1}).

To prove the first inequality in (\ref{eq:lem1_2_comb})
assume
(\ref{eq:pr_lem1_1}) holds
for some $j=2k$ with $k\in\mymathbb{N}$,
where $\genD_{\nu}\in\hat{\mathcal{R}} %{\delta}_{\myv{m}}$ if $\nu$ is odd
and $\genD_{\nu}\in\hat{\mathcal{R}} %{\delta}_{\myv{n}}$ otherwise.
Note that $\myv{n}=R_{\myv{l}}(\delta)\myv{m}$ and hence, for $U=R_{\myv{n}}(\alpha} %{\beta')R_{\myv{l}}(\beta} %{\gamma'+\delta)R_{\myv{m}}(\gamma} %{\delta')$ in Proposition~\ref{lem:const_even},
\[
d(U\myv{m},\myv{n})=d(R_{\myv{l}}(\beta} %{\gamma'+\delta)\myv{m},\myv{n})=d(R_{\myv{l}}(\beta} %{\gamma'+\delta)\myv{m},R_{\myv{l}}(\delta)\myv{m})=(\beta} %{\gamma'+\delta)-\delta=\beta} %{\gamma'.
\]
Then, we have
$\beta} %{\gamma' \le (2 k -1) \delta$ by (\ref{eq:lem3_2}) of Lemma~\ref{lem:ti}.
This implies $\lceil (\beta} %{\gamma'+\delta)/(2\delta) \rceil \le k$, and therefore,
\begin{equation}\label{eq:evenlb}
2\Big\lceil \frac{\beta} %{\gamma'+\delta}{2\delta} \Big\rceil \le 2k=j.
\end{equation}
From this bound,
we have
the first inequality in (\ref{eq:lem1_2_comb})
and hence, the equality among all sides of (\ref{eq:lem1_2_comb}).
This shows (\ref{eq:lem1_2})  in the case where $f(\alpha} %{\beta,\beta} %{\gamma,\delta) \ge \delta$. The proof of (\ref{eq:lem1_2}) in the other case is given in \refapp{app:case2}.
This completes the proof of the lemma.
The proved lemma immediately implies the corollary.
\hfill $\Box$\vspace*{1ex}
\iffalse
\mbox{} \hspace{1ex} \mbox{}
\hfill $\Box$} % \vspace*{1ex}
\fi

} %{\noindent%
{\em Proof of Theorem~\ref{th:num_rot}.}\/
Note that for any $U\in {\rm SU}(2)$,
\[
N_{\myv{m},\myv{n}}(U)
= \min \{
\varM{\myv{m},\myv{n}}{\rm odd}(U),
\varM{\myv{m},\myv{n}}{\rm even}(U),
\varM{\myv{n},\myv{m}}{\rm odd}(U),
\varM{\myv{n},\myv{m}}{\rm even}(U)
\} ,
\]
and we can write $U$ in terms of three parametric expressions:
\begin{equation*}
U=R_{\myv{u}}(\theta)=U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}=
U_{\tilde{\alpha} %{\beta},\tilde{\beta} %{\gamma},\tilde{\gamma} %{\delta}}^{\myv{n},-\myv{l}}
\end{equation*}
where $\beta} %{\gamma,\tilde{\beta} %{\gamma} \in [0, \pi]$, $\alpha} %{\beta,\gamma} %{\delta,\tilde{\alpha} %{\beta},\tilde{\gamma} %{\delta},\theta \in \mymathbb{R}$,
and $\myv{u}\in S^2$.
Then, we have
\[
\frac{\beta}{2}=\arcsin \bigg[ \sqrt{1-(\rinnpr{\myv{m}}{\myv{u}})^2}\Big|\sin\frac{\theta}{2}\Big| \bigg]
\quad\mbox{and}\quad
\frac{\tilde{\beta}}{2}=\arcsin \bigg[ \sqrt{1-(\rinnpr{\myv{n}}{\myv{u}})^2}\Big|\sin\frac{\theta}{2}\Big| \bigg]
\]
owing to Lemma~\ref{lem:EulerG}, and hence,
\[
\varM{\myv{m},\myv{n}}{\rm odd}(U)=
2\Big\lceil \frac{\arcsin \sqrt{1-(\rinnpr{\myv{m}}{\myv{u}})^2}\abs{\sin\mfrac{\theta}{2}}}{\delta} \Big\rceil +1
\]
and
\[
\varM{\myv{n},\myv{m}}{\rm odd}(U)=
2\Big\lceil \frac{\arcsin \sqrt{1-(\rinnpr{\myv{n}}{\myv{u}})^2}\abs{\sin\mfrac{\theta}{2}}}{\delta} \Big\rceil +1
\]
owing to Lemma~\ref{lem:1}. Then, if $\abs{\rinnpr{\myv{m}}{\myv{u}}} \ge \abs{\rinnpr{\myv{n}}{\myv{u}}}$
whenever $\sin (\theta/2) \ne 0$, which implies
$\varM{\myv{m},\myv{n}}{\rm odd}(U) \le \varM{\myv{n},\myv{m}}{\rm odd}(U)$, we shall have
\begin{eqnarray}
N_{\myv{m},\myv{n}}(U)
&= &\min \{
\varM{\myv{m},\myv{n}}{\rm odd}(U),
\varM{\myv{m},\myv{n}}{\rm even}(U),
\varM{\myv{n},\myv{m}}{\rm even}(U)
\} \notag\\
&= &\min \Big\{
2 \Big\lceil \frac{\beta}{2\delta} \Big\rceil +1,
g(\alpha} %{\beta,\beta} %{\gamma,\delta),
\varM{\myv{n},\myv{m}}{\rm even}(U)
 \Big\} \label{eq:lem1pre}
\end{eqnarray}
for $U=U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}$.
But $[\sin (\theta/2) \ne 0 \rightarrow \abs{\rinnpr{\myv{m}}{\myv{u}}} \ge \abs{\rinnpr{\myv{n}}{\myv{u}}}]$ follows from
$\myb(\myv{m},U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}}) \ge \myb(\myv{n},U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})$
by the definition of $\myb$.
[This is because writing $U$ in (\ref{eq:u123}) as $U=R_{\myv{u}}(\theta)$,
$\theta\in\mymathbb{R}$, $\myv{u}\in S^2$, results in
$-\sin(\theta/2) \myv{u} = (x,y,z)^{\rm T}$ as in Section~\refroyalsub{ss:prel}{sssub:para},
whereby $\myb(\myv{v},U)=\abs{\sin(\theta/2)} \abs{ \rinnpr{\myv{u}}{\myv{v}} }$.]
Hence,
we have (\ref{eq:lem1pre}).

A short additional argument (\refappcomma{app:tech}) shows
\begin{equation}
\varM{\myv{n},\myv{m}}{\rm even}(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})
= g(\gamma} %{\delta,-\beta} %{\gamma,\delta), \label{eq:a_even}
\end{equation}
and therefore,
\[
N_{\myv{m},\myv{n}}(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})
=
\min \Big\{ 2 \Big\lceil \frac{\beta}{2\delta} \Big\rceil +1 , \,
g(\alpha} %{\beta,\beta} %{\gamma,\delta) , \,
g(\gamma} %{\delta,-\beta} %{\gamma,\delta)
 \Big\}.
\]
Finally, from Corollary~\ref{coro:1} or from the argument in \refappcomma{app:N_mnUandD},
it readily follows that
$
N_{\myv{m},\myv{n}}\big(F(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})\big) =
N_{\myv{m},\myv{n}}(U_{\alpha} %{\beta,\beta} %{\gamma,\gamma} %{\delta}^{\myv{m},\myv{l}})
$.
Hence, we obtain
the theorem.
\hfill $\Box$\vspace*{1ex}

From the viewpoint of construction, we summarise the (most directly) suggested way to obtain an optimal construction of a given element $U\in {\rm SU}(2)$,
where we assume $\delta=\arccos\rinnpr{\myv{m}}{\myv{n}}\in (0,\pi/2]$
without loss of generality.
If $\myb(\myv{m},U)$ $\ge \myb(\myv{n},U)$,
choose a construction
that attains the minimum in (\ref{eq:lem1pre}). The construction is
among that of Proposition~\ref{lem:const_odd}, that of Proposition~\ref{lem:const_even}, and
that of Proposition~\ref{lem:const_even} applied to $U^{\dagger}$ in place of $U$
[note $U^{\dagger}=R_{\myv{u}_1}(\phi_1) \cdots R_{\myv{u}_j}(\phi_j)$ implies $U=R_{\myv{u}_j}(-\phi_j)\cdots R_{\myv{u}_1}(-\phi_1)$].
\begin{comment} OR
\end{comment}
If $\myb(\myv{m},U) < \myb(\myv{n},U)$,
interchanging $\myv{m}$ and $\myv{n}$, apply the construction just described.%
\footnote{One (seemingly difficult) issue arises:
Determine all optimal decompositions of an arbitrarily fixed rotation.
Note that in Propositions~\ref{lem:const_odd} and \ref{lem:const_even}
and their proofs,
any solution for
$R_{\myv{n}}(\theta) = R_{\myv{m}}(\alpha} %{\beta)R_{\myv{l}}(\beta} %{\gamma ) R_{\myv{m}}(\gamma} %{\delta)$
can be used (see Corollary~\ref{coro:8} in \refapp{app:proof_rem} for explicit solutions,
among which one is chosen to be used in Remarks~\ref{rem:const_expl} and \ref{rem:const_expl_even}).}

\section{Conclusion \label{ss:conc}}

This work has established
the least value $N_{\myv{m},\myv{n}}(U)$
of a positive integer $k$ such that $U$ can be decomposed into the product of $k$ rotations about either $\myv{m}$ or $\myv{\varvn}$
for an arbitrarily fixed
element $U$ in ${\rm SU(2)}$, or in ${\rm SO}(3)$, where $\myv{m},\myv{n} \in S^2$
are arbitrary
real unit vectors with $\abs{\rinnpr{\myv{m}}{\myv{n}}}<1$.
\iffalse
$N_{\myv{m},\myv{n}}(\cD) = \min \{ k \in \mymathbb{N} \mid \forall U \in \cD, \ N_{\myv{m},\myv{n}}(U) \le k \}$,

is obtained as
$
N_{\myv{m},\myv{n}}(\cD) = \lceil  \pi / \arccos \abs{\rinnpr{\myv{m}}{\myv{\varvn}}} \rceil +1
$ for any $\myv{m},\myv{\varvn}\in S^2$ with $\abs{\rinnpr{\myv{m}}{\myv{\varvn}}} < 1$, and for $\cD = {\rm SU}(2),{\rm SO}(3)$.
\fi

\section*{Acknowledgments}
This work was supported by SCOPE (Ministry of Internal Affairs and Communications), and by Japan Society for the Promotion of Science KAKENHI Grant numbers 22540150
and 21244007.

redpajama_set_name: RedPajamaArXiv


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Q: OSX Yosemite - JVM Shared library does not contain the JNI_CreateJavaVM symbol Downloaded OS X Yosemite Beta 3 and I'm trying to run some eclipse based IDE's, namely Titanium Studio and Aptana Studio.
Running into this error message when I try to start the apps. Didn't work on boot, so I downloaded JDK 1.8.0_40, still no dice. It's looking in the proper location, the file exists and it's using the correct java version. Not sure what the issue is, any help would be appreciated.

A: I ran into this problem as well. Apparently Aptana Studio is 32-bit, while the java I have installed is 64-bit.
I didn't want to revert to Java6 as the accepted answer suggested, so I solved it another way.
I downloaded a 64-bit version of Eclipse and installed the plugin version of Aptana Studio.

A: Download the newest version of Java for OS X 2014-001 from Apple. http://support.apple.com/kb/DL1572?viewlocale=en_US&locale=en_US
I encountered the same error this morning and this was a quick and easy fix.

A: I got this problem when I try to lunch AptanaStudio 3.1 on OSX yosemite.
I google this problem for quite a lot and found that JDK1.8(64-bit) version just does not support the Aptana Studio(3.1), maybe also not working on some other IDE which based on the same skeleton of Eclipse.
The only solution I can came out is to install an older version of JDK, and I found that the link @ChancePance provide just worked for me.
Why I am answering this question is just to specify that it's OK to install this version of JDK(which is version 1.6.0_65, 64-bit when I download the package) alongside with the J2SE 1.8(provided by Oracle). So you don't have to worry about there is any conflicts while installing two version of JDK.
I need the JDK1.8 for some of the new features of the latest version of Oracle's J2SE. And I need Aptana Studio at the same time. I am quite worried about there might be some conflicts.
Below is what I have done on OSX Yosemite to solve the problem of AptanaStudio3.1:

*

*Download and install the package: JavaForOSX.

*Found the file "AptanaStidio3.ini", the file by default located under the the directory here: /Applications/Aptana Studio 3/AptanaStudio3.app/Contents/MacOS

*Add "-vm" argument to specify the java which will be used to lunch Aptana Studio.
-vm
/Library/Java/JavaVirtualMachines/1.6.0.jdk/Contents/Home/bin/java
Hope this will help you guys who got the same problem.

A: I was also experiencing this problem because I had a 64 bit JDK 1.7 installed (the standard Oracle version - not the Apple version indicated in the other answers) and had mistakenly downloaded a 32 bit version of Eclipse. Doh!!
I removed the 32 bit version and downloaded the 64 bit version of Eclipse and everything works great now.

A: Simply downloading the newest Java for OSX worked the charm. I clicked and downloaded the file from the link above -> http://support.apple.com/kb/DL1572?viewlocale=en_US&locale=en_US <

A: Open "/Applications/BonitaBPMCommunity-6.5.0/BonitaBPMCommunity.app/Contents/MacOS/BonitaBPMCommunity.ini"
Obtain the location of the JRE / JDK on your system by executing in a terminal:
/usr/libexec/java_home
Add 2 lines after adapting the ...jdk1.8.0... to your system as found out above

-vm
/Library/Java/JavaVirtualMachines/jdk1.8.0_31.jdk/Contents/Home/bin/java
in between the following lines:

-clearPersistedState
-vmargs

redpajama_set_name: RedPajamaStackExchange


text:
U.S. Air Force Hints at Secret Spy Drone
March 10, 2020 Topic: Security Blog Brand: The Buzz Tags: U.S. Air ForceDroneMilitaryTechnology
The U.S. Air Force is asking Congress to let it retire nearly 150 older warplanes in order to free up money for new technology, all starting with the service's 2021 budget. The idea immediately ran into a wall of resistance from lawmakers. Now the flying branch is trying to make the case that the new technology is worth the cuts.
The U.S. Air Force is asking Congress to let it retire nearly 150 older warplanes in order to free up money for new technology, all starting with the service's 2021 budget.
The idea immediately ran into a wall of resistance from lawmakers. Now the flying branch is trying to make the case that the new technology is worth the cuts.
But there's a problem. Much of the new tech is classified, the Air Force has claimed. So it can't talk about it in public.
The impasse raises several important questions. How secret is too secret? And what exactly is the Air Force developing that justifies reductions in systems that, until recently, the service wanted more of, not fewer?
The Air Force is asking Congress for $170 billion for 2021, compared to the $166 billion that the service got in 2020. Flattening spending is compelling the Air Force and sister services to reconsider the mix of new and existing systems they want to pay for.
The Air Force decided it prefers to risk short-term cuts if that means it might be able to accelerate long-term modernization. To that end, the Air Force's 2021 budget proposal retires 44 A-10 attack planes, 29 KC-135 and KC-10 tankers, 24 C-130H airlifters, 24 E/RQ-4 spy drones and 17 B-1B bombers.
The proposed cuts have startled some observers because, as recently as 2018, the Air Force was telling lawmakers it wanted to massively expand its force structure from 312 front-line squadrons to 386.
That expansion would cost tens of billions of dollars and require hundreds if not thousands of new aircraft on top of the roughly 5,500 planes the service currently has. The expansion plan in particular called for many more drones, tankers and bombers.
But it's possible the Air Force wants to use the savings from the proposed 2021 retirements to acquire secret planes types, effectively preserving if not expanding existing force structure.
Air Force chief of staff Gen. David Goldfein more or less said that during March 2020 testimony before a U.S. House of Representatives committee.
"Not surprisingly, of the services, the air and space forces have the largest classified portfolio of investment," Goldfein said. "This makes the story harder to tell."
Goldfein told lawmakers he could brief them in a classified setting. "Since most of what we're retiring is unclassified and visible, while many of our game-changing investments are classified and therefore invisible, we want to thank many of you for taking our classified briefing and offer it to any of you, or your staff, between now and end-game. When you see what we're trading for, our budget submission will make perfect sense."
Common sense dictates that the secret technology Goldfein alluded to includes at least one new drone type. After all, the Air Force operates, but declines to discuss, a small number of RQ-180 stealth spy drones -- and reportedly is working to expand the fleet.
The RQ-180's existence could help to explain why the Air Force wants to retire most of its roughly 40 E/RQ-4 drones, which on average are less than a decade old.
The high-flying but non-stealthy RQ-4 has proved to be vulnerable to enemy air defenses. Iranian forces in the summer of 2019 shot down a Navy RQ-4 variant over the Strait of Hormuz.
But while it could seem reasonable for the Air Force to cut RQ-4s in favor of a similar number of new and better RQ-180s, cuts to bombers and tankers might make less sense. After all, the service almost definitely does not possess secret tanker and bomber types.
The new KC-46 tanker and even newer B-21 bomber both are public programs of record. The KC-46 has suffered serious developmental problems. Flaws in its refueling system mean the Air Force can't safely use them as tankers in anything short of a national emergency.
The B-21 meanwhile is still in development and probably won't fly for the first time until the end of 2021.
So Goldfein's argument -- that plus-ups for secret tech make up for reductions in non-secret tech -- only goes so far. Near-term cuts to tankers and bombers really are cuts. And if Congress lets the Air Force start retiring some older planes in 2021, in many cases the service will just have to make do with fewer planes.
Even though Air Force leaders have said for years they need more, not fewer, of exactly these plane types.
David Axe serves as Defense Editor of the National Interest. He is the author of the graphic novels War Fix, War Is Boring and Machete Squad.

redpajama_set_name: RedPajamaCommonCrawl


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Kalinino is a village in the Goygol Rayon of Azerbaijan. It is named after Mikhail Kalinin, a Soviet revolutionary and a member of the Politburo of the Communist Party of the Soviet Union.

References

Populated places in Goygol District

redpajama_set_name: RedPajamaWikipedia


text:
Hoppegarten är en förortskommun öster om Berlin i Tyskland, belägen i Landkreis Märkisch-Oderland i förbundslandet Brandenburg.  Kommunen bildades 2003 genom sammanslagning av de tidigare kommunerna Dahlwitz-Hoppegarten, Hönow och Münchehofe.

Namnet Hoppegarten betyder ungefär "Humlegården" och syftar på odling av växten humle.  Kommunen är mest känd för sin galoppbana, Galopprennbahn Hoppegarten, grundad 1868.

Geografi
Kommunen är en del av Storstadsregion Berlin/Brandenburg och är den västligast belägna kommunen i Landkreis Märkisch-Oderland, omedelbart öster om Berlins stadsgräns och cirka 20 km öster om Berlins centrum.

Sedan sammanslagningen 2003 är kommunens område uppdelat i två delar, som skiljs åt av Berlinstadsdelen Mahlsdorf; norr om Mahlsdorf ligger kommundelen Hönow, söder om Mahlsdorf ligger kommundelarna Dahlwitz-Hoppegarten och Münchehofe.

Källor

Kommuner och städer i Landkreis Märkisch-Oderland


In order to study and compare the correlation between the quantitative and qualitative traits of sugar beet genotypes, using multivariate methods under Rhizoctonia disease a trial was carried out at Miandoab Agricultural and Natural Resources Station in 2014.The experimental design was randomized complete block design with four replications. The experimental materials were sixteen sugar beet genotypes. Analysis of variance showed significant differences among the genotypes for all traits, except root sodium content. Based on stepwise regression analysis, gross sugar yield, molasses sugar content and gross sugar content by justifying %89.99 of changes in white sugar yield were considered as the most effective traits in justification of white sugar yield. Factor analysis identified five factors by justifying 81.62% of total data variances. Finally, the three groups were set up in the cluster analysis at similarity distance of 10. The cluster 2 was subjected to groups a (the highest rate) for yellowing score, uniformity score, growth score, healthy plant number, root yield, gross sugar yield, white sugar content, white sugar yield and extract percentage, and was placed in group b (the lowest rate) for root potassium content and alkalinity. Eventually, based on path analysis and regression analysis, the gross sugar yield was determined as the most effective trait on the white sugar yield and the genotypes in cluster 2 were identified suitable for cultivation under the conditions of Rhizoctonia infestation.


Power up with Proverbs: Cometh the hour cometh the man.
"It is always darkest before the dawn" says one well known proverb and in times of crisis we look for a capable leader. Nobody suitable seems to be available but when the situation reaches critical it can stir the sleeping abilities of their hero to burst forth and shine. The ordinary seeming individual will find strengths within and rise to the challenge. Such as William Wallace, Oliver Cromwell, George Washington or Mahatma Gandhi.


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WASHINGTON, Jan. 22 (Xinhua) -- It seems surreal but in fact real: U.S. President Donald Trump spends his first full day in office on Saturday denying his feud with the CIA, ignoring the massive protests against him, and blasting the media over lessening his inaugural crowd.
The fights greatly overshadowed the new president's three diplomatic moves announced in the day: to eradicate "radical Islamic terrorism," to host British Prime Minister Theresa May as first visiting foreign leader on Thursday, and to meet with his Mexican counterpart Enrique Pena Nieto on Jan. 31.
At the CIA headquarters in Langley, Virginia, Trump described radical Islamic terrorism as "evil" unseen before and should be "eradicated just off the face of the Earth."
"We gonna end it. It's time. It's time right now to end it," Trump told hundreds of CIA staff, offering no details of his plan.
However, public attention was diverted when he quickly turned to attack the U.S. media over their reports about his inaugural crowd size and his feud with the intelligence community.
At the CIA headquarters, he scolded the U.S. media as "the most dishonest people" by reporting his feud with the intelligence agencies, although he previously slapped them in public for leaking reports that Russia may have "compromising materials" about him.
He further accused the media of lying about the number of people attending his inauguration on Friday. Earlier estimates put the crowd gathered for Trump's inauguration at about one-third the size of Obama's in 2009.
"I get up this morning and I turn on one of the networks and they show an empty field. I said wait a minute, I made a speech, I looked out, the field was, it looked like a million, a million and a half people. They showed a field where there were practically nobody standing there," he told the CIA staff.
"So, we caught" the media, Trump said, "and we caught them in beauty. And I think they're going to pay a big price."
Hours later, White House spokesman Sean Spicer used his first official appearance in the press briefing room to criticize the U.S. media for the coverage of Trump's inauguration crowd, blasting their reports on the crowd size as "shameful and wrong."
"This was the largest audience to ever witness an inauguration, both in person and around the globe," Spicer said, despite photos and video showing much fewer people assemble on the National Mall for Trump's inauguration than for Obama's in 2009.
Photos comparing the two inaugurations side-by-side on social media were "intentionally framed in a way to minimize support" for the new president, said Spicer. He did not take questions from reporters.
Both Trump and his spokesman have so far remained silent to the widespread anti-Trump protests held across the United States and parts of the world on Saturday.
Large turnouts were reported in Washington, Boston, New York, Chicago and Los Angeles in a powerful display of rejection of Trump's agenda, and of what the organizers said his insulting attitudes toward women. Similar marches were held in Paris, London, Sydney and some other major cities in the West.
More than 2.5 million Americans joined the protests across the country, reported TheHill news daily, which cited a review of official and unofficial estimates from U.S. largest cities.
In New York, Bridget Mills, a 41-year-old protester, told Xinhua: "I hope it sends a message (to Trump) that we're listening, we are paying attention to if he cares morally about people's civil rights and America's standing in the world."
Elizabeth Moore, a protester from New Haven, Connecticut, said: "I am very, very concerned about the next four years. I think we have a lot of rights that are going to be trampled on under the Trump administration."
"His bigotry and his views toward women are extremely troubling to me," she added.
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text:
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Airbus Deutschland Gmbh v. Shahram Kolahzadeh
The Complainant is Airbus Deutschland Gmbh, Hamburg, Germany, represented by Bird & Bird Solicitors, France.
The Respondent is Shahram Kolahzadeh, Grampian, United Kingdom of Great Britain and Northern Ireland.
The disputed domain names are <airbusa340.com>, <airbus320.com>, <airbus321.com>, <airbus380.com> and <flyairbus.com>.
The registrar with which all the domain names are registered is Wild West Domains, Inc.
The Complaint was filed with the WIPO Arbitration and Mediation Center (the "Center") on January 23, 2007. On January 25, 2007, the Center transmitted by email to Wild West Domains, Inc. a request for registrar verification in connection with the domain names at issue. On the same day, Wild West Domains, Inc. transmitted by email to the Center its verification response confirming that the Respondent is listed as the registrant of the disputed domain names and providing the contact details for the administrative, billing, and technical contact. The Center verified that the Complaint satisfied the formal requirements of the Uniform Domain Name Dispute Resolution Policy (the "Policy"), the Rules for Uniform Domain Name Dispute Resolution Policy (the "Rules"), and the WIPO Supplemental Rules for Uniform Domain Name Dispute Resolution Policy (the "Supplemental Rules").
In accordance with the Rules, paragraphs 2(a) and 4(a), the Center formally notified the Respondent of the Complaint, and the proceedings commenced on January 31, 2007.
Upon the Complainant's requests on February 20 and March 9, 2007 respectively, the Proceeding was suspended from February 21, 2007 until April 13, 2007.
In accordance with the Rules, paragraph 5(a), the due date for Response was April 24, 2007. The Response was filed with the Center on April 24, 2007.
On May 2, 2007, the Complainant filed a supplemental submission with the Center.
The Center appointed Dr. Bernhard F. Meyer-Hauser as the sole panelist in this matter on May 22, 2007. The Panel finds that it was properly constituted. The Panel has submitted the Statement of Acceptance and Declaration of Impartiality and Independence, as required by the Center to ensure compliance with the Rules, paragraph 7.
The Complainant belongs to the Airbus group which was established in 1970 as a European consortium of French, German and later Spanish and United Kingdom companies. In 2001, the consortium became a fully integrated company, incorporated as a simplified joint stock company or S.A.S (Sociйtй par Actions Simplifiйe) under French law ("Airbus"). Since 2006, the European Aeronautic Defence and Space Company (EADS) owns 100% of the Airbus shares.
Airbus group is one of the two leading aircraft manufacturers in the world and takes around half of all aircraft orders for airliners with more than 100 seats. Airbus group has more than 150 sites throughout the world, including 16 development and manufacturing facilities in France, Germany (where the Complainant is established), United Kingdom (where Respondent is living) and Spain, where Airbus aircrafts are designed, built, assembled and tested. Airbus also operates several subsidiaries, regional offices, training centers, spare centers and resident customer services offices. In total, Airbus has 55,000 employees worldwide.
The Complainant has filed trademark applications for AIRBUS, A320, A321, A340 and A380 in more than 40 countries, including the European Union, United States of America, China and Taiwan, Province of China. The numbers reflected in Airbus' registered trademarks correspond to so-called Airbus aircraft families, whereby the designation A320 (and its subcategories A318, A319 and A321) are used for relatively small, single-aisle aircrafts while the designations A330 and A340 are used for relatively large, wide-body, long distance planes. At the top of the Airbus product range is the double-deck A380 which is not yet commercially operating.
The disputed Airbus domain names were registered by the Respondent at the following dates:
<airbus320.com> on October 9, 2003;
<airbus321.com> on May 17, 2004;
<airbusa340.com> on August 2, 2005;
<airbus380.com> on January 25, 2004;
<flyairbus.com> on March 13, 2005.
The websites associated with the domain names are currently inactive.
5. Parties' Contentions
The Complainant objects to the use of the disputed domain names by the Respondent and bases its Complaint on the following grounds:
1. The disputed domain names are all identical or confusingly similar to the trademarks in which the Complainant has rights.
The Respondent's registrations are combinations or modifications of the Complainant's trademarks AIRBUS, A320, A321, A340 and A380 with the adjunction of the gTLD ".com".
The domain names <airbus320.com>, <airbus321.com>, <airbusa340.com> and <airbus380.com> incorporate two of the Complainant's trademarks. Even if the disputed domain names are not exactly identical to the registered trademarks, it is clear that they are confusingly similar since each of them contains distinctive elements of the Complainant's relevant trademarks.
Furthermore, the sole fact of modifying the Complainant's trademarks by removing the letter "A" before 320, 321 and 380 for the domain names <airbus320.com>, <airbus321.com> and <airbus380.com> does not allay the similarity to the Complainant's trademarks.
Regarding the domain name <flyairbus.com>, it shall to be considered that the mere adding of a generic word to the trademark does not prevent the finding of confusing similarity between the domain name and the trademark.
2. The Respondent has no legitimate interest in respect of the disputed domain names.
The Respondent has never been authorized by the Complainant to use the domain names <airbus320.com>, <airbus321.com>, <airbusa340.com>, <airbus380.com> and <flyairbus.com>: The Complainant has prior rights in the trademarks AIRBUS, A380, A320, A321 and A340, which were all registered before the Respondent registered the disputed domain names.
Furthermore, the Respondent must have registered the disputed domain names with the sole intention to divert consumers to its own websites and to prevent the Complainant from reflecting its trademarks on the corresponding domain names. This behaviour cannot be regarded as a bona fide offering of goods or services or a fair use of the domain name.
3. Finally, the Respondent registered (and uses) the domain names in bad faith for the following reasons:
The Respondent must have been aware that the Complainant is one of the two leading aircraft manufacturers worldwide. The Respondent registered the disputed domain names in order to prevent the Complainant from reflecting the marks on the corresponding websites and to disrupt the Complainant's business.
When the Complainant sent a cease and desist letter to the Respondent on July 21, 2006, the contentious domain names were leading to a portal promoting links to travels. Since then, the disputed domain names lead to an error webpage. The mere holding of an inactive website must also be considered as a registration and use in bad faith.
The Respondent alleges that he is entitled to use the disputed domain names for the following reasons:
1. The domain names are not identical or confusingly similar to a trademark or a service mark in which the Complainant has rights.
The domain names <airbus380.com>, <airbus320.com>, <airbus321.com> and <flyairbus.com> are combinations of a number or a word and "airbus", a name which happens to be trademarked. These domain names are comparable to <driveford.com>, <buyBMW.com> or <airbusflight.com>, which are all active sites on the Internet.
The Respondent acknowledges that the Complainant has rights in the marks AIRBUS, A380, A320, A321 and A340. However, this does not include any rights in the specific numbers or in the combination of numbers and word "airbus".
2. The Respondent has a legitimate interest in using the domain name.
The Respondent was not aware of the existence of the Complainant's above mentioned trademarks until he received the cease and desist letter on July 21, 2006.
The disputed domain names were initially linked to a web construction page for a forum site. After the Respondent had received the cease and desist letter, the links to the original web page were suspended for the duration of the proceeding. This was communicated to the Complainant on July 27, 2006.
The domain name was registered in order to provide a common site for aviation enthusiast and pilots to discuss aviation topics and recent developments, such as other existing sites like <airbusa380.com>.
Moreover, the Respondent has neither made any profit from the disputed domain names, nor has he tried to sell them for profit. The Respondent, therefore, never intended and does not intend to misleadingly divert consumers for commercial gain.
3. The Respondent did not register the domain names in bad faith for the following reasons:
The Respondent has never tried to rent, sell or lease any of the disputed domain names to the Complainant, its competitors or any third party. Furthermore, the Respondent has not drawn any profit from the registration or use of the domain names.
Finally, the Respondent is not a competitor of the Complainant and did not register the domain names for the purpose of disrupting the Complainant's business. The sole purpose of acquiring the disputed domain names was to provide links to a discussion forum.
Procedural Matters
The Panel notes that the Center received a further submission by the Complainant on May 3, 2007. This submission had not been requested by the Panel pursuant to Rule 12.
Regarding supplemental submissions, Rule 12 of the Rules unambiguously provides that the Panel may request, in its sole discretion, further statements or documents from either of the parties. There is no provision in the Rules allowing a party to file an additional submission without leave of the Panel. This is because under the expedited process provided under the Policy and Rules each party is given one opportunity to put forward all the material on which it wishes to rely and is expected to do so (Delikomat Betriebsverpflegung Gesellschaft m.b.H. v. Alexander Lehner, WIPO Case No. D2001-1447).
After consideration of Complainant's supplemental submission, the Panel concludes that it does not address topics which Complainant could not have brought up in its original Complaint and the Panel decides not admit the supplemental filing.
Substantive Matters
Paragraph 4(a) of the Policy sets forth three elements that must be established by the Complainant to merit a finding that the Respondent has engaged in abusive domain name registration, and to obtain relief. These elements are that:
(i) Respondent's domain name is identical or confusingly similar to a trademark or service mark in which the Complainant has rights; and
(ii) Respondent has no rights or legitimate interests in respect of the domain name; and
(iii) Respondent's domain name has been registered and is being used in bad faith.
The domain name <airbusa340 > is a combination of two of the Complainant's registered trademarks (AIRBUS and A340). Thus, it is, if not identical, at least confusingly similar to the trademarks in which the Complainant has rights.
The domain names <airbus320.com>, <airbus321.com> and <airbus380.com> contain the Complainant's trademark AIRBUS and the numbers (or Airbus family names) 320, 321 and 380 respectively. These Airbus family names are, in turn, protected by the Complainant's trademarks A320, A321 and A380. The mere omission of the letter "A" next to the numbers (which obviously stands for "Airbus") does not allay the confusing similarity of the disputed domain names to the Complainant's trademarks.
The domain name <flyairbus.com> combines the Complainant's trademark AIRBUS with the generic term "fly". Neither the addition of an ordinary descriptive word (either as prefix or suffix) nor the suffix ".com" detract from the overall impression and the dominant part of the name is instantly recognizable as the famous trademark AIRBUS. The addition of the word "fly" to the Complainant's trademark even increases the likelihood of confusion since Airbus is commonly known as a major aircraft manufacturer that obviously wishes the world to fly its aircrafts (see Sony Kabushiki Kaisha (also trading as Sony Corporation) v. Inja, Kil, WIPO Case No. D2000-1409; UBS AG v. Rudolf Gautsch, WIPO Case No. D2005-0993).
The Panel finds that the Complainant therefore has fulfilled the requirements of paragraph 4(a)(i) of the Policy.
According to paragraph 4(c) of the Policy, the following circumstances can demonstrate rights to and legitimate interests of a Respondent in a domain name:
(i) before any notice was given to the Respondent of the dispute, the Respondent used, or demonstrably made preparations to use, the domain name or a name corresponding to the Domain Name in connection with a bona fide offering of goods or services; or
(ii) the Respondent (as an individual, business, or other organization) has been commonly known by the domain name, even though it has not acquired any trademark or service mark rights; or
(iii) The Respondent is making a legitimate non-commercial or fair use of the domain name.
The Complainant states that the Respondent was not authorised by the Complainant to use its trademarks nor any of the disputed domain names, and there is no evidence before the Panel that the Respondent has been commonly known by the disputed domain names.
The websites associated with the domain names currently show no content. While it appears that they were active before the Complainant sent a cease and desist letter to the Respondent on July 21, 2006, the facts concerning the previous and intended use of the domain names remain disputed and can not be verified by the Panel. The Complainant alleges that the corresponding websites were leading to a portal promoting links for travels, whereas the Respondent contends that the sole purpose of the registration was to provide a discussion forum for aviation enthusiasts without making any commercial use of the domain names.
In any case, the Respondent has not provided the Panel with sufficient information that would suggest that he had previously used the domain names for, or that he was making demonstrable preparations for future use in connection with, a bona fide offering of goods or services. In fact, it seems inconceivable that the Respondent could make any active use of the disputed domain names without creating at least an initial false impression of association with the Complainant.
The fact that the Respondent registered five different domain names, all of which are confusingly similar to the Complainant's trademarks, casts additional doubts on Respondent's purported intention to create a common site for aviation enthusiasts.
For all these reasons, the Panel finds that the Complainant has established a prima facie case, which the Respondent has failed to convincingly rebut, that the Respondent has no legitimate interest in respect of the domain names in the sense of paragraph 4(c) of the Policy.
According to paragraph 4(b) of the Policy, a domain name has been used and registered in bad faith if:
(i) circumstances indicating that Respondent has registered and has acquired the domain name primarily for the purpose of selling, renting, or otherwise transferring the domain name registration to the complainant who is the owner of the trademark or service mark or to a competitor of that complainant, for valuable consideration in excess of Respondent's documented out-of-pocket costs directly related to the domain name; or
(ii) Respondent has registered the domain name in order to prevent the owner of the trademark or service mark from reflecting the mark in a corresponding domain name, provided that Respondent has engaged in a pattern of such conduct; or
(iii) Respondent has registered the domain name primarily for the purpose of disrupting the business of a competitor; or
(iv) by using the domain name, Respondent has intentionally attempted to attract, for commercial gain, Internet users to its website or other on-line location, by creating a likelihood of confusion with the complainant's mark as to the source, sponsorship, affiliation, or endorsement of Respondent's website or location or of a product or service on its website or location.
The Respondent registered several domain names reflecting the famous trademark AIRBUS although he obviously never had any rights in this trademark. In the Panel's opinion, the Respondent was aware of the Complainant's trademark already at the time when he filed the registrations. The registration of five domain names incorporating the famous trademark AIRBUS, combined with additional numbers or prefixes that are themselves Airbus trademarks and which are strongly associated with aircraft types manufactured by the Airbus group, indicates that the purpose of the registrations was to prevent the owner of the trademarks from reflecting the marks in corresponding domain names. The registration of several names corresponding to Complainant's trademarks is sufficient itself to constitute a pattern of such conduct, and constitutes bad faith within the meaning of paragraph 4(b)(ii) of the Policy (General Electric Company v. Normina Anstalt a/k/a Igor Fyodorov, WIPO Case No. D2000-0452).
The Panel further notes that the Respondent has not provided any compelling explanation for the lack of discussion-related content on the websites associated with the domain names, or indeed the number of domain names registered. In the circumstances, the Panel is not persuaded by the Respondent's contention that the sole purpose of acquiring the various disputed domain names was to post links to AIRBUS discussion sites; a commercial or monetary motivation appears to the Panel to be a more likely explanation.
To summarise, the Respondent has registered and is using the domain names in bad faith in accordance with paragraph 4(b) of the Policy.
In consideration of the above, the Panel holds that the domain names <airbusa340.com>, <airbus320.com>, <airbus321.com>, <airbus380.com> and <flyairbus.com> are confusingly similar to the registered trademarks of the Complainant, that the Respondent has no rights or legitimate interests in respect of the domain names, and that the domain names have been registered and are being used in bad faith.
For all the foregoing reasons, in accordance with paragraphs 4(i) of the Policy and paragraph 15 of the Rules, the Panel orders that the domain names <airbusa340.com>, <airbus320.com>, <airbus321.com>, <airbus380.com> and <flyairbus.com> shall be transferred to the Complainant.
Dr. Bernhard F. Meyer-Hauser
Dated: June 3, 2007
Источник информации: https://www.internet-law.ru/intlaw/udrp/2007/d2007-0088.html


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cusati media GmbH
The agency - online marketing and e-commerce from a single source
How do you recognize good agencies with conscientious employees, a serious way of working and a trend-oriented sense for new developments? We are sure: not to elaborate image brochures, pompous TV creations or funny Flash animations - at least not necessarily. Our figureheads are on the one hand our diverse and satisfied customers from different branches and on the other hand our employees, whose competence is rounded off by their will to complete projects customer-friendly and in the best possible way.
Agency experience in online marketing from Web 1.0 to Web 2.1
Our agency experience starts at the end of the 90s. At that time, we remember, nobody could imagine the extent of today's Internet boom. At the beginning of the millennium, institutions and private individuals increasingly used the opportunity to present themselves online with the World Wide Web. But with this so-called Web 1.0 still, so much air remained up. The 2000s opened an unprecedented revolutionary market - suddenly everyone could present, share and market on the Internet. The emergence of countless online shops, communities, dating portals and chat rooms were the consequence. But as soon as they were created, there were many changes in the search engine ranking at the same time, in the offers of technologies and the interfaces of online & offline. Anyone who did not ask for an "up-to-date" mentality here had little chance of survival in the internet confused jungle; also because of the enormous competitive density. Keep in mind: Nowadays, everyone can use the power of the Internet for their own purposes.
Phone Number: +49.351.46925657
Website: https://www.cusatimedia.de
Bamberger Strasse 1, Dresden, , United States 01187
Revenue: $1 million
cusati media GmbH - Listing in January 2023
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