text:
\begin{document}
\maketitle
\begin{abstract}
We define in the space of $n \times m$ matrices of rank $n$, $n\leq m$, the condition Riemannian structure as follows: For a given matrix $A$ the tangent space at $A$ is equipped with the Hermitian inner product obtained by multiplying the usual Frobenius inner product by the inverse of the square of the smallest singular value of $A$ denoted $\si_n(A)$.
When this smallest singular value has multiplicity $1$, the function $A \rightarrow \log (\si_n(A)^{-2})$ is a convex function with respect to the condition Riemannian structure that is $t \rightarrow \log (\si_n(A(t))^{-2})$ is convex, in the usual sense for any geodesic $A(t)$.
In a more abstract setting, a function $\alpha$ defined on a Riemannian manifold $(\CM , \left\langle ,\right\rangle)$ is said to be self-convex when $\log \alpha (\gamma(t))$ is convex for any geodesic in $(\CM , \al \left\langle , \right\rangle)$. Necessary and sufficient conditions for self-convexity are given when $\alpha$ is $C^2$. When $\alpha(x) = d(x,\CN)^{-2}$ where $d(x,\CN)$ is the distance from $x$ to a $C^2$ submanifold  $\CN \subset \R^j$ we prove that $\alpha$ is self-convex when restricted to the largest open set of  points $x$ where there is a unique closest point in $\CN$ to $x$. We also show,  using this more general notion, that the square of the condition number $\left\| A \right\|_F/ \sigma_n(A)$ is self-convex in projective space and the solution variety.
\end{abstract}

\section{Introduction}

Let two integers $1\leq n\leq m$ be given and let us consider the space of matrices $\Knm$, $\K = \R$ or $\C$, equipped with the Frobenius Hermitian product
$$\left\langle M,N \right\rangle_F = \trace (N^*M) = \sum_{i,j}m_{ij}\overline{n_{ij}}.$$
Given an absolutly continuous path $A(t)$, $a \le t \le b$, its length is given by the integral
$$L = \int_a^b \left\|\frac{dA(t)}{dt}\right\|_F dt,$$
and the shortest path connecting $A(a)$ to $A(b)$ is the segment connecting them. Consider now the problem of connecting these two matrices with the shortest possible path in staying, as much as possible, away from the set of ``singular matrices'' that is the matrices with non-maximal rank.

The singular values of a matrix $A \in \Knm$ are denoted in non-increasing order:
$$\si_1(A) \ge \ldots \ge \si_{n-1}(A) \ge \si_n(A) \ge 0.$$
We denote by $\GL$ the space of matrices $A \in \Knm$ with maximal rank : $\rank A = n$, that is $\si_n(A) > 0$ so that the set of singular matrices is
$$\CN = \Knm \setminus \GL = \left\{ A \in \Knm \ : \ \si_n(A) = 0 \right\} .$$
Since the smallest singular value of a matrix is equal to the distance from the set of singular matrices:
$$\si_n(A) = d_F(A, \CN) = \min_{S \in \CN} \left\| A-S \right\|_F,$$
given an absolutly continuous path $A(t)$, $a \le t \le b$, we define its ``condition length'' by the integral
$$L_\ka = \int_a^b \left\|\frac{dA(t)}{dt}\right\|_F \si_n(A(t))^{-1}dt.$$
A good compromise between length and distance to $\CN$ is obtained in minimizing $L_\ka$. We call  ``minimizing condition geodesic'' an absolutly continuous path, parametrized by arc length,  which minimizes $L_\ka$ in the set of absolutly continuous paths with given end-points and condition distance $d_\ka(A,B)$ between two matrices the length $L_\ka$ of a minimizing condition geodesic with endpoints $A$ and $B$, if any.

In this paper our objective is to investigate the properties of the smallest singular value $\sigma_n(A(t))$ along a condition geodesic. Our main result says that the map $\log \left(\sigma_n(A(t))^{-1}\right)$ is convex. Thus $\sigma_n(A(t))$ is concave, and its minimum value along the path is reached at one of the endpoints.

Note that a similar property holds in the case of hyperbolic geometry where instead of $\Knm$ we take $\R^{n-1} \times [0, \infty[$, instead of $\CN$ we have
$\R^{n-1} \times \left\{ 0 \right\}$, and where the length of a path $a(t) = (a_1(t), \ldots , a_n(t))$ is defined by the integral
$$\int \left\|\frac{da(t)}{dt}\right\| a_n(t)^{-1}dt.$$
Geodesics in that case are arcs of circles centered at $\R^{n-1} \times \left\{0\right\}$ or segments of vertical lines, and $\log \left( a_n(t)^{-1} \right)$ is convex along such paths.

The approach used here to prove our theorems is heavily based on Riemannian geometry. We define on $\GL$ the following Riemannian structure:
$$\left\langle M,N \right\rangle_{\ka, A} = \si_n(A)^{-2} {\rm Re}\left\langle M,N \right\rangle_F$$
where $M, N \in \Knm$ and $A \in \GL$. The minimizing condition geodesics defined previously are clearly geodesic in $\GL$ for this Riemannian structure so that we may use the toolbox of Riemannian geometry. In fact things are not so simple: the smallest singular value $\sigma_n(A)$ is a locally Lipschitz map in $\GL$, and it is smooth on the open subset
$$\GLS = \left\{ A \in \GL \ : \ \sigma_{n-1}(A) > \sigma_{n}(A)\right\}$$
that is when the smallest singular value of $A$ is simple. On the open subset $\GLS$ the metric $\left\langle \cdot , \cdot \right\rangle_\ka$ defines a smooth Riemannian structure, and we call ``condition geodesics'' the geodesics related to this structure. Such a path is not necessarily a minimizing geodesic. Our first main theorem establishes a remarkable property of the condition Riemannian structure:

\begin{theorem} \label{th-1} $\si_n^{-2}$ is logarithmically convex on $\GLS$ i.e. for any geodesic curve $\ga(t)$ in $\GLS$ for the condition metric the map
$\log \left(\si_n^{-2} (\ga(t))\right)$ is convex.
\end{theorem}

\begin{problem}\label{prob-1}
The condition Riemannian structure $\left\langle .,. \right\rangle_\ka$ is defined in $\GL$ where it is is only locally Lipschitz. Let us define condition geodesics in $\GL$ as the extremals of the condition length $L_\ka$ (see for example
\cite{cla} Chapter 4, Theorem 4.4.3, for the definition of such extremals in the Lipschitz case).
Is Theorem \ref{th-1} still true for $\GL$? All the examples we have studied confirm that convexity holds, even if $\si_n^{-1} (\ga(t))$ fails to be $C^1$. See Boito-Dedieu \cite{boi}. We intend to address this issue in a future paper.
\end{problem}

In a second step we extend these results to other spaces of matrices: the sphere $\S_r(\GLS )$ of radius $r$ in $\GLS$ in Corollary \ref{cor-3b}, the projective space $\P\left( \GLS \right)$ in Corollary \ref{cor-3c}. We also consider the case of the solution variety of the homogeneous equation
$M \zeta = 0$ that is the set of pairs
$$\left\{(M, \zeta) \in {\mathbb K}^{n \times (n+1)} \times {\mathbb K}^{n + 1} \ : \ M \zeta = 0 \right\} .$$
Now our function $\al$ is the square of the condition number studied by Demmel in \cite{dem}.
This is done in the affine context in Theorem \ref{th-3} and in the projective context in Corollary \ref{cor-4}.

\vskip 3mm Since $\si_n(A)$ is equal to the distance from $A$ to the set of singular matrices a natural question is to ask whether our main result remains valid for the inverse of the distance from certain sets or for more general functions.

\begin{definition} \label{def-3} Let $(\CM,\langle\cdot,\cdot\rangle)$ be Riemannian and let $\al : \CM \rightarrow \R$ be a function of class $C^2$ with positive values. Let $\CM_\ka$ be the manifold $\CM$ with the new metric
$$\langle\cdot,\cdot\rangle_{\ka , x} = \al (x)\langle\cdot,\cdot\rangle_x $$
called condition Riemann structure.
We say that $\al$ is {\it self-convex} when $\log \al(\ga (t))$ is convex for any geodesic $\ga$ in  $\CM_\ka$.
\end{definition}

For example, with $\CM = \left\{ x=(x_1, \ldots , x_n) \in \R^n \ : \ x_n>0 \right\}$ equipped with the usual metric, $\al(x) = x_n^{-2}$ is self-convex. The space $\CM_\ka$ is the Poincar\'e model of hyperbolic space.

In the following theorem we prove self-convexity for the distance function to a $C^2$ submanifold without boundary $\CN \subset \R^j$. Let us denote by
$$\rho(x) = d(x,\CN) = \min_{y \in \CN} \left\| x-y \right\| \ \mbox{and} \ \al(x) = \frac{1}{\rho(x)^2}.$$
Let $\CU$ be the largest open set in $\R^j$ such that, for any $x \in \CU$, there is a unique closest point in $\CN$ to $x$. When $\CU$ is equipped with the new metric $\al(x)\left\langle .,. \right\rangle$ we have:

\begin{theorem} \label{th-2} The function $\al : \CU \setminus \CN \rightarrow \R$ is self-convex.
\end{theorem}

Theorem \ref{th-2} is then extended to the projective case. Let $\CN$ be a $\mathcal{C}^2$ submanifold without boundary of $\mathbb{P}(\mathbb{R}^j)$.
Let us denote by $d_R$ the Riemannian distance in projective space (points in the projective space are lines throught the origin and the distance $d_R$ between two lines is the angle they make). Let us denote $d_\mathbb{P}=\sin d_R$ (this is also a distance), define $\al(x)=d_\mathbb{P}(x,\CN)^{-2}$, and let $\CU$ be the largest open subset of $\mathbb{P}(\mathbb{R}^j)$ such that for $x\in \CU$ there is a unique closest point from $\CN$ to $x$ for the distance $d_\P$. Then

\begin{corollary}\label{cor-th-2}
The map $\al : \CU \setminus \CN \rightarrow \mathbb{R}$ is self-convex.
\end{corollary}

The extension of Theorem \ref{th-1} and Theorem \ref{th-2} to other types of sets or functions is not obvious. In Example \ref{ex-Frob} we prove that
$\al(A)=\sigma_1(A)^{-2}+\cdots+\sigma_n(A)^{-2}$ is not self-convex in $\GL$.

In Example \ref{ex-circle} we take $\CN = \R^2$, and $\CU$ the unit disk so that $\CU$ contains a point (the center) which has many closest points from $\CN$. In that case the corresponding function $\al : \CU \setminus \CN \rightarrow \R$ is self-convex but it fails to be smooth at the center of the disk.

In Example \ref{ex-two-points} we provide an example of a submanifold $\CN \subset \R^2$ such that the function $\al(x) = d(x, \CN)^{-2}$ defined on  $\R^2 \setminus \CN$ is not self-convex.

Our interest in considering the condition metric in the space of matrices comes from recent papers by Shub \cite{shu} and Beltr\'an-Shub \cite{bel} where these authors use condition length along a path in certain solution varieties to estimate step size for continuation methods to follow these paths. They give bounds on the number of steps required in terms of the condition length of the path. If geodesics in the condition metric are followed the known bounds on polynomial system solving are vastly improved. To understand the properties of these geodesics we have begun in this paper with linear systems where we can investigate their properties more deeply. We find self-convexity in the context of this paper remarkable. We do not know if similar issues may naturally arise in linear algebra even for solving systems of linear equations. Similar issues do clearly arise when studying continuation methods for the eigenvalue problem.

\section{Self-convexity}

Let us first start to recall some basic definitions about convexity on Riemannian manifolds. A good reference on this subject is
Udri\c ste \cite{udr}.

\begin{definition} \label{def-2} We say that a function $f : \CM \rightarrow \R$ is {\it convex} whenever
$$f(\ga_{xy}(t)) \leq (1-t)f(x) + t f(y)$$
for every $x, y \in \CM$, for every geodesic arc $\ga_{xy}$ joigning $x$ and $y$ and $0 \le t \le 1$.
\end{definition}

The convexity of $f$ in $\CM$ is equivalent to the convexity in the usual sense of $f \circ \ga_{xy}$ on $[0,1]$ for every $x,y \in U$ and the geodesic $\ga_{xy}$ joining $x$ and $y$ or also to the convexity of $g \circ \ga$  for every geodesic $\ga$ (\cite{udr} Chap. 3, Th. 2.2). Thus, we see that

\begin{lemma} Self-convexity of a function $\al : \CM \ra \R$ is equivalent to the convexity of $\log \circ \al$ in the condition Riemannian manifold $\CM_\ka$.
\end{lemma}

\vskip 3mm When $f$ is a function of class $C^2$ in the Riemannian manifold $\CM$, we define its second derivative $D^2 f(x)$ as the second covariant derivative. It is a symmetric bilinear form on $T_x \CM$. Note (\cite[Chapter 1]{udr}) that if $x\in\CM$ and $\dot x\in T_x\CM$, and if $\gamma(t)$ is a geodesic in $\CM$, $\gamma(0)=x,\frac{d}{dt}\gamma(0)=\dot x$, then
\[
D^2f(x)(\dot x,\dot x)=\frac{d^2}{dt^2}(f\circ\gamma)(0).
\]
This second derivative depends on the Riemannian connection on $\CM$. Since $\CM$ is equipped with two different metrics: $\left\langle .,. \right\rangle$ and $\left\langle .,. \right\rangle_\ka$ we have to distinguish between the corresponding second derivatives; they are denoted by $D^2f(x)$ and $D_\ka^2f(x)$ respectively. No such distinction is necessary for the first derivative $Df(x)$.

Convexity on Riemannian manifold is characterized by (see \cite{udr} Chap. 3, Th. 6.2):

\begin{proposition}\label{pro-1}
A function $f : \CM \rightarrow \R$ of class $C^2$ is convex if and only if $D^2f(x)$ is positive semidefinite for every $x \in \CM$.
\end{proposition}

We use this proposition to obtain a caracterisation of self-convexity: $\al$ is self-convex if and only if the second derivative $D_\ka^2(\log \circ \al)(x)$ is positive semidefinite for any $x \in \CM_\ka$. We get

\begin{proposition}\label{pro-2}
For a function $\al : \CM \rightarrow \R$ of class $C^2$ with positive values self-convexity is equivalent to
\[
2\al(x)D^2\al(x)(\x,\x) + \|D\al(x)\|_x^2  \|\x\|_x^2 - 4(D\al(x)\x)^2 \geq 0
\]
for any $x\in \CM$ and for any vector $\x \in T_x\CM$, the tangent space at $x$.
\end{proposition}

\begin{proof}
Let $x \in \CM$ be given. Let $\varphi : \R^m \rightarrow \CM$ be a coordinate system such that $\varphi(0)=x$ and with first fundamental form $g_{ij}(0)=\delta_{ij}$ (Kronecker's delta) and Christoffel's symbols $\Gamma_{jk}^i(0) = 0$, and let
$$A = \al \circ \varphi $$
so that $\al(x) = A(0)$. Those coordinates are called ``normal'' or ``geodesic''. Note that this implies
\[
\frac{\partial g_{ij}}{\partial z_k}(0)=0
\]
for all $i,j,k$. We denote by $g_{\ka,ij}$ and $\Gamma_{\ka,jk}^i$ respectively the first fundamental form and the Christoffel symbols for $\varphi$ in $\CM_\ka$. Let us compute them. Note that
\[
g_{\ka,ij}(z)=g_{ij}(z)A(z),
\]
\[
\frac{\partial g_{\ka,ij}}{\partial z_k}(0) = Dg_{\ka,ij}(0)(e_k) = D(g_{ij}A)(0)(e_k) =
\]
\[
g_{ij}(0)DA(0)(e_k) + A(0)Dg_{ij}(0)(e_k) = \delta_{ij}\frac{\partial A}{\partial z_k}(0).
\]
Moreover,
\[
\Gamma_{\ka,jk}^i = \frac{1}{A(0)} \Gamma_{jk}^i = \frac{1}{2A(0)}\left(\frac{\partial g_{\ka,ij}}{\partial z_k}(0) + \frac{\partial g_{\ka,ik}}{\partial z_j}(0) -
\frac{\partial g_{\ka,jk}}{\partial z_i}(0)\right) =
\]
\[
\frac{1}{2A(0)}\left(\delta_{ij}\frac{\partial A}{\partial z_k}(0) + \delta_{ik}\frac{\partial A}{\partial z_j}(0) -
\delta_{jk}\frac{\partial A}{\partial z_i}(0)\right).
\]
That is,
\[
\begin{cases}
\Gamma_{\ka, ik}^i = \Gamma_{\ka, ki}^i = \frac{1}{2A(0)}\frac{\partial A}{\partial z_k}(0)&\mbox{for all } i,k,\\
\Gamma_{\ka, jj}^i = \frac{-1}{2A(0)}\frac{\partial A}{\partial z_i}(0),&j\neq i,\\
\Gamma_{\ka, jk}^i = 0&\mbox{otherwise}.
\end{cases}
\]
The second derivative of the composition of two maps
$$\CM \stackrel{f}{\rightarrow} \R \stackrel{\psi}{\rightarrow} \R$$
is given by the identity (see \cite{udr} Chap. 1.3, Hessian)
$$D^2(\psi \circ f)(x) = D\psi(f(x))D^2f(x) + \psi''(f(x))Df(x)\otimes Df(x)$$
and where $Df(x)\otimes Df(x)$ is the bilinear form on $T_x \CM $ by
$$(Df(x)\otimes Df(x))(u,v) = Df(x)(u) Df(x)(v).$$
This gives in our context, that is when $f = \al$ and $\psi = \log$,
$$D_\ka^2(\log \circ \al)(x) = \frac{1}{\al(x)}D_\ka^2 \al(x) - \frac{1}{\al(x)^2}D\al(x)\otimes D\al(x).$$

\vskip 3mm According to Proposition \ref{pro-1} our objective is now to give a necessary and sufficient condition for $D_\ka^2(\log \circ \al)(x)$ to be positive semidefinite for each $x \in \CM$.
In our system of local coordinates the components of $D^2\al(x)$ are (see \cite{udr} Chap. 1.3)
$$A_{jk} = \frac{\partial^2A}{\partial z_j \partial z_k} - \sum_i \Gamma_{jk}^i \frac{\partial A}{\partial z_i} = \frac{\partial^2A}{\partial z_j \partial z_k}$$
while the components of $D_\ka^2\al(x)$ are
$$A_{\ka,jk} = \frac{\partial^2A}{\partial z_j \partial z_k} - \sum_i \Gamma_{\ka,jk}^i \frac{\partial A}{\partial z_i}.$$
If we replace the Christoffel symbols in this last sum by the values previously computed we obtain, when $j=k$,
$$\sum_i \Gamma_{\ka,jj}^i \frac{\partial A}{\partial z_i} = \Gamma_{\ka,jj}^j \frac{\partial A}{\partial z_j} +
\sum_{i \ne j} \Gamma_{\ka,jj}^i \frac{\partial A}{\partial z_i} = $$
$$\frac{1}{2A}\left( \frac{\partial A}{\partial z_j} \right)^2 - \frac{1}{2A}\sum_{i \ne j}\left( \frac{\partial A}{\partial z_i} \right)^2 = \frac{1}{A}\left( \frac{\partial A}{\partial z_j} \right)^2 - \frac{1}{2A}\sum_{i}\left( \frac{\partial A}{\partial z_i} \right)^2$$
while when $j \ne k$,
$$\sum_i \Gamma_{\ka,jk}^i \frac{\partial A}{\partial z_i} = \Gamma_{\ka,jk}^j \frac{\partial A}{\partial z_j} +
\Gamma_{\ka,jk}^k \frac{\partial A}{\partial z_k} = $$
$$\frac{1}{2A} \frac{\partial A}{\partial z_k} \frac{\partial A}{\partial z_j} + \frac{1}{2A} \frac{\partial A}{\partial z_j} \frac{\partial A}{\partial z_k} = \frac{1}{A} \frac{\partial A}{\partial z_j} \frac{\partial A}{\partial z_k}.$$
Both cases are subsumed in the identity
$$\sum_i \Gamma_{\ka,jk}^i \frac{\partial A}{\partial z_i} = \frac{1}{A} \frac{\partial A}{\partial z_j} \frac{\partial A}{\partial z_k} - \frac{\de_{jk}}{2A}\sum_{i}\left( \frac{\partial A}{\partial z_i} \right)^2.$$
Putting together all these identities gives the following expression for the components of $D_\ka^2(\log \circ \al)(x)$:
$$D_ka^2(\log \circ \al)(x)_{jk} = \frac{1}{A} \left(\frac{\partial^2A}{\partial z_j \partial z_k} -  \frac{1}{A} \frac{\partial A}{\partial z_j} \frac{\partial A}{\partial z_k} + \frac{\de_{jk}}{2A} \sum_i \left( \frac{\partial A}{\partial z_i} \right)^2\right)
- \frac{1}{A^2}\frac{\partial A}{\partial z_j} \frac{\partial A}{\partial z_k} =$$
$$\frac{1}{2A^2}\left(2A\frac{\partial^2A}{\partial z_j \partial z_k} + \de_{jk} \sum_i \left( \frac{\partial A}{\partial z_i} \right)^2 - 4\frac{\partial A}{\partial z_j} \frac{\partial A}{\partial z_k}\right)
.$$
Thus, $D_\ka^2(\log \circ \al)(x) \ge 0$ if and only if
$$2\al(x)D^2\al(x) + \left\| D\al(x) \right\|_x^2\left\langle .,. \right\rangle_x - 4D\al(x)\otimes D\al(x)$$
is positive semi-definite, that is when
\[
2\al(x)D^2\al(x)(\x,\x) + \|D\al(x)\|_x^2 \|\x\|_x^2 - 4(D\al(x)\x)^2 \geq 0
\]
for any $x\in \CM$ and for any vector $\x \in T_x\CM$. This finishes the proof.
\end{proof}

\vskip 3mm An easy consequence of Proposition \ref{pro-2} is the following. See also Example \ref{ex-two-points}.

\begin{corollary}\label{cor-0}
When a function $\al : \CM \rightarrow \R$ of class $C^2$ is self-convex then any critical point of $\al$ has a positive semi-definite second derivative $D^2\al(x)$. Such a function cannot have a strict local maximum or a non-degenerate saddle.
\end{corollary}

\begin{proposition}\label{pro-3}
The following condition is equivalent for a $C^2$ function $\al=1/\rho^2 : \CM \longrightarrow \R$ to be self-convex on $\CM$: For every $x \in \CM$ and $\x\in T_x \CM$,
\[
\|\x\|^2\|D\rho(x)\|^2-(D\rho(x)\x)^2-\rho(x)D^2\rho(x)(\x,\x)\geq0,
\]
or, what is the same,
\[
2\|\x\|^2\|D\rho(x)\|^2\geq D^2\rho^2(x)(\x,\x).
\]
\end{proposition}

\begin{proof}
Note that
\[
D\al(x)\x=\frac{-2}{\rho(x)^3}D\rho(x)\x,
\]
\[
D^2\al(x)(\x,\x)=\frac{6}{\rho(x)^4}(D\rho(x)\x)^2-\frac{2}{\rho(x)^3}D^2\rho(x)(\x,\x).
\]
Hence, the necessary and sufficient condition of Proposition \ref{pro-2} reads
\[
\frac{4\|\x\|^2\|D\rho(x)\|^2}{\rho(x)^6}-\frac{16}{\rho(x)^6}(D\rho(x)\x)^2+\frac{12}{\rho(x)^6}(D\rho(x)\x)^2-\frac{4}{\rho(x)^5}D^2\rho(x)(\x,\x)\geq0,
\]
and the proposition follows.
\end{proof}

\begin{corollary}\label{cor-1}
Each of the following conditions is sufficient for a function $\al = 1/\rho^2:\CM\longrightarrow\R$ to be self-convex at $x\in \CM$: For every $\x\in T_x\CM$,
\[
D^2\rho(x)(\x,\x)\leq0,
\]
or
\[
\|D^2\rho^2(x)\|\leq2\|D\rho(x)\|^2.
\]
\end{corollary}

In the following proposition we obtain a weaker condition on $\al$ to obtain convexity in $\CM_\ka$ instead of self-convexity.

\begin{proposition}\label{prop:convexnolog}
$\al(x)$ is convex in $\CM_\ka$ if and only if
\[
2\al(x)D^2\al(x)(\x,\x) + \|D\al(x)\|_x^2\|\x\|_x^2 - 2(D\al(x)\x)^2\geq0,
\]
for any $x\in\CM$ and any vector $\x \in T_x\CM$.
\end{proposition}
\begin{proof}
We follow the lines of the proof of Proposition \ref{pro-2} with $\psi$ equal to the identity map instead of $\psi = \log$.
\end{proof}

\section{Some general formulas for matrices}

\begin{proposition}\label{pro-4}
Let $A = (\Si, 0) \in \GLS$, where $\Si = \diag (\si_1\geq\cdots\geq \si_{n-1}>\si_n) \in \K^{n \times n}$. The map $\si_n : \GLS \rightarrow \R$ is a smooth map and, for every $U \in \K^{n \times m}$,
$$\left\{
\begin{array}[pos]{l}
	D\si_n(A)U = \mathrm{Re}(u_{nn}),\\
	D^2\sigma^2_n(A)(U,U) = 2\sum_{j=1}^m |u_{nj}|^2 - 2\sum_{k=1}^{n-1}\frac{|u_{kn}\si_n + \overline{u_{nk}}\si_k|^2}{\si_k^2-\si_n^2}.
\end{array}
\right.
$$
\end{proposition}

\begin{proof} Since $\si_n^2$ is an eigenvalue of $AA^*$ with multiplicity $1$, the implicit function theorem proves the existence of smooth functions $\si^2_n(B) \in \R$ and $u(B) \in \K^n$, defined in an open neighborhood of $A$ and satisfying
$$\left\{
\begin{array}[pos]{l}
	BB^*u(B) = \si^2_n(B)u(B),\\
	\left\| u(B) \right\|^2 = 1,\\
	u(A) = e_n = (0, \ldots ,0,1)^T \in \K^n,\\
	\si^2_n(A) = \si_n^2.
\end{array}
\right.
$$
Differentiating these equations at $B$ gives, for any $U \in \K^{n \times m}$,
$$\left\{
\begin{array}[pos]{l}
	(UB^*+BU^*)u(B) + BB^*\u(B)= \left( {\sigma_n^2} \right)' u(B) + \sic(B) \u (B),\\
	u(B)^*\u(B) = 0
\end{array}
\right.
$$
with $\u (B) = Du(B)U$ and $ \left( {\sigma_n^2} \right)' = D\sic (B)U$. Pre-multiplying the first equation by $u(B)^*$ gives
$$u(B)^*(UB^*+BU^*)u(B) + u(B)^*BB^*\u(B)=  \left( {\sigma_n^2} \right)' u(B)^*u(B) + \sic(B) u(B)^* \u (B)$$
so that
$$D\sic (B)U =  \left( {\sigma_n^2} \right)' = 2 \mathrm{Re} (u(B)^* UB^* u(B))$$
and
$$D \si_n(B)U = \frac{\mathrm{Re} (u(B)^* UB^* u(B))}{\si_n(B)}.$$
The derivative of the eigenvector is now easy to compute:
$$D u(B)U = \u (B) = (\sic (B) I_n - BB^*)^\dagger (UB^*+BU^* -  \left( {\sigma_n^2} \right)' I_n)u(B)$$
where $(\sic (B) I_n - BB^*)^\dagger$ denotes the generalized inverse (or Moore-Penrose inverse) of $\sic (B) I_n - BB^*$.

The second derivative of $\sic$ at $B$ is given by
$$D^2\sic(B)(U,U) = 2 \mathrm{Re} (\u(B)^*UB^* u(B) + u(B)^* UU^* u(B) + u(B)^*UB^* \u(B)) = $$
$$2 \mathrm{Re} (u(B)^* UU^* u(B) + u(B)^*(UB^*+BU^*) \u(B)) = 2 \mathrm{Re} (u(B)^* UU^* u(B) + $$
$$u(B)^*(UB^*+BU^*)(\sic (B) I_n - BB^*)^\dagger (UB^*+BU^* -  \left( {\sigma_n^2} \right)' I_n)u(B)).$$
Using $u(A) = e_n$ and $\si_n(A) = \si_n$ we get
$$\left\{
\begin{array}[pos]{l}
	D\sic (A)U = 2 \mathrm{Re} (UA^*)_{nn} = 2 \si_n \mathrm{Re}( u_{nn}),\\
	D\si_n (A)U = \mathrm{Re}( u_{nn}),\\
\end{array}
\right.
$$
and the second derivative is given by
$$D^2\sic(A)(U,U) = $$
$$2 \mathrm{Re} \left( (UU^*)_{nn} + \sum_{k=1}^{n-1}(UA^*+AU^*)_{nk}(\si_n^2 - \si_k^2)^{-1}(UA^*+AU^* -  \left( {\sigma_n^2} \right)' I_n)_{kn} \right) = $$
$$2 \mathrm{Re} \left( (UU^*)_{nn} + \sum_{k=1}^{n-1}\frac{\left|(UA^*+AU^*)_{kn}\right|^2}{\si_n^2 - \si_k^2} \right) =
2 \sum_{j=1}^{m} \left| u_{nj} \right|^2 - 2 \sum_{k=1}^{n-1}\frac{\left| u_{kn}\si_n + \overline{u_{nk}}\si_k \right|^2}{\si_k^2 - \si_n^2}.$$
\end{proof}

\begin{corollary}\label{cor-2}
Let $A = (\Si, 0) \in \GLS$, where $\Si = \diag(\si_1\geq\cdots\geq \si_{n-1}>\si_n>0) \in \K^{n \times n}$. Let us define $\rho(A) = \si_n(A) / \left\| A \right\|_F$. Then, for any $U \in \K^{n \times m}$ such that $\mathrm{Re} \left\langle A,U \right\rangle_F = 0$, we have
$$\left\{
\begin{array}[pos]{l}
	D\rho (A)U = \frac{\mathrm{Re}( u_{nn})}{\left\| A \right\|_F},\\
	D^2\rho^2 (A)(U,U) = \frac{2}{\left\| A \right\|_F^2}\left(\sum_{j=1}^m |u_{nj}|^2 - \sum_{k=1}^{n-1}\frac{|u_{kn}\si_n + \overline{u_{nk}}\si_k|^2}{\si_k^2-\si_n^2} - \frac{\left\| U \right\|_F^2}{\left\| A \right\|_F^2}\si_n^2\right).\\
\end{array}
\right.
$$
\end{corollary}
\begin{proof}
Note that
\[
D\rho(A)U = \frac{D\sigma_n(A)U \|A\|_F - \sigma_n(A)\frac{2\mathrm{Re}\langle A,U\rangle_F}{2\| A \|_F}}{\| A \|_F^2}=\frac{D\sigma_n(A)U}{\| A \|_F},
\]
and the first assertion of the corollary follows from Proposition \ref{pro-4}.
For the second one, note that $h={h_1}/{h_2}$ (for real valued $\mathcal{C}^2$ functions $h,h_1,h_2$ with $h_2(0)\neq0$) implies
\[
D^2h = \frac{h_2^2D^2h_1 - h_1h_2D^2h_2 - 2h_2Dh_1Dh_2 + 2h_1(Dh_2)^2}{h_2^3}.
\]
Now, $\rho^2(A)={\sigma_n^2(A)}/{\| A \|_F^2}$, $D(\| A \|_F^2)U = 2\mathrm{Re}\langle A,U \rangle_F = 0,$ $D^2(\| A \|_F^2)(U,U) = 2\| U \|_F^2,$
and $D^2\sigma_n^2(A)(U,U)$ is known from Proposition \ref{pro-4}. The formula for $D^2\rho^2 (A)$ follows after some elementary calculations.
\end{proof}

\section{The affine linear case}

We consider here the Riemannian manifold $\CM = \GLS$ equipped with the usual Frobenius Hermitian product.
Let $\al :\GLS \rightarrow \R$ be defined as $\al(A) = {1}/{\sigma_n^2(A)}$.

\begin{corollary}\label{cor-3} The function $\al$ is self-convex in $\GLS$.
\end{corollary}

\begin{proof}
From Proposition \ref{pro-3}, it suffices to see that
\[
2\| U \|_F^2 \| D\sigma_n(A) \|_F^2  \geq D^2\sigma_n^2(A)(U,U).
\]
Since unitary transformations are isometries in $\GLS$ with respect to the condition metric we may suppose, via a singular value decomposition that $A = (\Si, 0) \in \GLS$, where $\Si = \diag(\si_1 \geq \cdots \geq \si_{n-1} > \si_n) \in \K^{n \times n}$. Now, the inequality to verify is obvious from Proposition \ref{pro-4}, as $\| D\sigma_n(A) \|_F = 1$ and
\[
D^2\sigma_n^2(A)(U,U) = 2 \sum_{j=1}^m|u_{nj}|^2 - 2\sum_{k=1}^{n-1}\frac{| u_{kn}\si_n + \overline{u_{nk}}\si_k |^2}{\si_k^2 - \si_n^2} \leq 2\sum_{j=1}^m |u_{nj}|^2 \leq 2 \| U \|_F^2.
\]
\end{proof}

\begin{corollary}\label{cor-3b} Let $r>0$. The function $\al$ is
self-convex in the sphere $\S_r(\GLS)$ of radius $r$ in $\GLS$.
\end{corollary}

\begin{proof}
It is enough to prove that any geodesic in $(\S_r(\GLS), \al)$ is also
a geodesic in $(\GLS, \al)$. Indeed, suppose that $A$ and $B$ are
matrices in $\S_r(\GLS)$ and the minimal geodesic in $(\GLS, \al)$
between $A$ and $B$ is $X(t)$, $a \le t \le b$. Then we claim that
$L_\ka\left(\frac{rX(t)}{\left\| X(t) \right\|_F}\right) \le
L_\ka(X(t))$. Indeed, for any $t$,
$$\frac{d}{dt}\left(\frac{rX(t)}{\left\| X(t) \right\|_F}\right) =
\frac{r\frac{dX(t)}{dt}}{\left\| X(t) \right\|_F} -
r\frac{X(t)Re(\langle X(t),\frac{dX(t)}{dt}\rangle_F)}{\left\| X(t)
\right\|_F^3}$$
so that
$$
\left\|\frac{d}{dt}\left(\frac{rX(t)}{\|X(t)\|}\right)\right\|_F=
$$
$$
\left(\frac{r^2\left\|\frac{dX(t)}{dt} \right\|_F^2}{\left\| X(t)
\right\|_F^2}+\frac{r^2Re(\langle
X(t),\frac{dX(t)}{dt}\rangle_F)^2}{\left\| X(t)
\right\|_F^4}-\frac{2r^2Re(\langle
X(t),\frac{dX(t)}{dt}\rangle_F)^2}{\left\| X(t)
\right\|_F^4}\right)^{1/2}=
$$
$$
\left(\frac{r^2\left\|\frac{dX(t)}{dt} \right\|_F^2}{\left\| X(t)
\right\|_F^2}-\frac{r^2Re(\langle
X(t),\frac{dX(t)}{dt}\rangle_F)^2}{\left\| X(t)
\right\|_F^4}\right)^{1/2}\leq \frac{r\left\|\frac{dX(t)}{dt}
\right\|_F}{\left\| X(t) \right\|_F}
$$
Hence,
$$\left\| \frac{d}{dt}\left(\frac{rX(t)}{\left\| X(t)
\right\|_F}\right) \right\|_\ka =
\si_n^{-1}\left(\frac{rX(t)}{\left\| X(t) \right\|_F}\right)
\left\|\frac{d}{dt}\left(\frac{rX(t)}{\|X(t)\|}\right)\right\|_F=
$$
$$\frac{\|X(t)\|_F\si_n^{-1}\left(X(t)\right)}{r}
\left\|\frac{d}{dt}\left(\frac{rX(t)}{\|X(t)\|}\right)\right\|_F\le
\si_n^{-1}\left(X(t)\right)\left\| \frac{dX(t)}{dt} \right\|_F=
\left\| \frac{dX(t)}{dt} \right\|_\ka .$$
Therefore $X(t)$ can only be a minimizing geodesic if it belongs to
$\S_r(\GLS)$. Since all geodesics are locally minimizing geodesics,
Corollary \ref{cor-3b} follows.
\end{proof}

\vskip 3mm The following gives an example of a smooth and non-selfconvex function in $\GL$.

\begin{example}\label{ex-noconvexmuf}\label{ex-Frob}
For $n \ge 3$, the function $\al(A)=\sigma_1(A)^{-2}+\cdots+\sigma_n(A)^{-2}$ is not self-convex in $\GL$.
\end{example}
\begin{proof}
For simplicity we consider the case of real square matrices. We have $\al(A)=\|A^{-1}\|_F^2$,
\[
D\al(A)\dot{A}=-2\langle A^{-1},A^{-1}\dot{A}A^{-1}\rangle_F=-2\langle A^{-T}A^{-1}A^{-T},\dot{A}\rangle_F,
\]
\[
\|D\al(A)\|_F^2=4\|A^{-T}A^{-1}A^{-T}\|^2_F,
\]
\[
D^2\al(A)(\dot{A},\dot{A})=2\|A^{-1}\dot{A}A^{-1}\|^2_F+4\langle A^{-1},A^{-1}\dot{A}A^{-1}\dot{A}A^{-1}\rangle_F.
\]
According to Proposition \ref{prop:convexnolog}, the self-convexity of $\al(A)$ in $\mathbb{GL}_n$ is equivalent to
\[
2\|A^{-1}\|_F^2\left(2\|A^{-1}\dot{A}A^{-1}\|^2_F+4\langle A^{-1},A^{-1}\dot{A}A^{-1}\dot{A}A^{-1}\rangle_F\right)+
\]
\[
4\|\dot{A}\|_F^2\|A^{-T}A^{-1}A^{-T}\|^2_F-8\langle A^{-1},A^{-1}\dot{A}A^{-1}\rangle_F^2\geq0
\]
This inequality is not satisfied when
\[
A=\begin{pmatrix}1&0&0\\0&1&0\\0&0&2\end{pmatrix}\text{   and   }\dot{A}=\begin{pmatrix}0&1&0\\-1&0&0\\0&0&0\end{pmatrix}.
\]
\end{proof}

\section{The homogeneous linear case}

\subsection{The complex projective space.}

The matter of this subsection is mainly taken from Gallot-Hulin-Lafontaine \cite{gal} sect. 2.A.5.

Let $V$ be a Hermitian space of complex dimension $\dim_\C V = d+1$. We denote by $\P(V)$ the corresponding projective space that is the quotient of $V \setminus \left\{ 0 \right\}$ by the group $\C^*$ of dilations of $V$; $\P(V)$ is equipped with its usual smooth manifold structure with complex dimension $\dim \P(V) = d$. We denote by $p$ the canonical surjection.

Let $V$ be considered as a real vector space of dimension $\dim_\R V = 2d+2$ equipped with the scalar product $\mathrm{Re}\left\langle .,. \right\rangle_V$.
The sphere $\S(V)$ is a submanifold in $V$ of real dimension $2d+1$. This sphere being equipped with the induced metric becomes a Riemannian manifold and, as usual, we identify the tangent space at $z \in \S(V)$ with
$$T_z \S(V) = \left\{ u \in V \ : \ \mathrm{Re}\left\langle u,z \right\rangle_V = 0 \right\}.$$

The projective space $\P(V)$ can also be seen as the quotient $\S(V) / S^1$ of the unit sphere in $V$ by the unit circle in $\C$ for the action given by $(\la , z) \in S^1 \times \S(V) \rightarrow \la z \in \S(V)$. The canonical map is denoted by
$$p_V : \S(V) \rightarrow \P(V).$$
$p_V$ is the restriction of $p$ to $\S(V)$.

The horizontal space at $z \in \S(V)$ related to $p_V$ is defined as the (real) orthogonal complement of $\ker Dp_V(z)$ in $T_z \S(V)$. This horizontal space is denoted by $H_z$. Since $V$ is decomposed in the (real) orthogonal sum
$$ V = \R z \oplus \R iz \oplus z^\perp $$
and since $\ker Dp_V(z) = \R iz$ (the tangent space at $z$ to the circle $S^1 z$) we get
$$H_z  = z^\perp = \left\{ u \in V \ : \ \left\langle u,z \right\rangle = 0 \right\}.$$

There exists on $\P(V)$ a unique Riemannian metric such that $p_V$ is a Riemannian submersion that is, $p_V$ is a smooth
submersion and, for any $z \in \S(V)$, $Dp_V(z)$ is an isometry between $H_z$ and $T_{p(z)}\P(V)$. Thus, for this Riemannian structure, one has:
$$\left\langle Dp_V(z)u, Dp_V(z)v \right\rangle_{T_{p(z)}\P(V)} = \mathrm{Re} \left\langle u,v \right\rangle_V$$
for any $z \in \S(V)$ and $u,v \in H_z$.

\begin{proposition} \label{pro-CP} Let $z \in \S(V)$ be given.
\begin{enumerate}
\item A chart at $p(z) \in \P(V)$ is defined by
$$\varphi_z : H_z \rightarrow \P(V), \ \ \varphi_z(u) = p(z+u).$$

\item Its derivative at $0$ is the restriction of $Dp(z)$ at $H_z$:
$$D\varphi_z (0) = Dp(z) : H_z \rightarrow T_{p(z)}\P(V)$$
which is an isometry.

\item For any smooth mapping $\psi : \P(V) \rightarrow \R$, and for any  $v \in H_z$ we have
$$D\psi(p(z))\left(Dp(z) v\right) = D(\psi \circ \varphi_z)(0)v$$
and
$$D^2\psi(p(z))(Dp(z)v, Dp(z)v) = D^2(\psi \circ \varphi_z)(0)(v,v).$$
\end{enumerate}
\end{proposition}

\begin{proof} 1 and 2 are easy. We have $D(\psi \circ \varphi_z)(0) = D\psi(p(z))D(\varphi_z)(0)$ which gives 3 since
$D(\varphi_z)(0)v = Dp(z)v$ for any $v \in H_z$. For the second derivative, recall that $D^2\psi(p(z))(Dp(z)v,Dp(z)v)=(\psi\circ\tilde{\gamma})''(0)$, where $\tilde{\gamma}$ is a geodesic curve in $\P(V)$ such that $\tilde{\gamma}(0)=p(z),\tilde{\gamma}'(0)=Dp(z)v$. Now, consider the horizontal $p_V-$lift $\gamma$ of $\tilde{\gamma}$ to $\S(V)$ with base point $z$. Note that $\gamma(0)=z,\gamma'(0)=v$. Hence,
\[
(\psi\circ\tilde{\gamma})''(0)=(\psi\circ p\circ \gamma)''(0)=
D^2(\psi\circ p)(z)(v,v)+D\psi(p(z))Dp(z)\gamma''(0).
\]
As $\gamma''(0)$ is orthogonal to $T_z \S(V)$, we have $Dp(z)\gamma''(0)=0$. Finally,
\[
D^2(\psi\circ p)(z)(v,v)=(\psi\circ p (z+tv))''(0)=(\psi\circ\varphi_z(tv))''(0)=D^2(\psi\circ\varphi_z)(0)(v,v),
\]
and the assertion on the second derivative follows.

\end{proof}

The following result will be helpful.

\begin{proposition}\label{pro-RS}
Let $\CM_1,{\CM}_2$ be Riemannian manifolds and $\al_2:\CM_2 \rightarrow ]0,\infty[$ be of class ${C}^2$. Let $\pi:\CM_1 \rightarrow \CM_2$ be a Riemannian submersion. Let $\CU_2 \subseteq \CM_2$ be an open set and assume that $\al_1 = \al_2 \circ \pi$ is self-convex in $\CU_1 = \pi^{-1}(\CU_2)$. Then, $\al_2$ is self-convex in $\CU_2$.
\end{proposition}

\begin{proof}
Let $\CM_{\ka,1}$ be $\CM_1$, but endowed with the condition metric given by $\al_1$, and let $\CM_{\ka,2}$ be $\CM_2$, but endowed with the condition metric given by $\al_2$. Then, $\pi : \CM_{\ka,1} \rightarrow \CM_{\ka,2}$ is also a Riemannian submersion.

Now, let $\gamma_2 : [a,b] \rightarrow \CU_2 \subseteq \CM_{\ka,2}$ be a geodesic, and let $\gamma_1 \subseteq \CM_{\ka,1}$ be its horizontal lift by $\pi$. Then, $\gamma_1$ is a geodesic in $\CU_1 \subseteq \CM_1$ (see \cite[Cor 2.109]{gal}) and hence $\log \al_1(\ga_1(t))$ is a convex function of $t$. Now,
\[
\log(\al_2(\gamma_2(t))) = \log(\al_2 \circ \pi(\ga_1 (t))) = \log(\al_1(\ga(t))),
\]
is convex as wanted. \end{proof}

\begin{corollary}\label{cor-3c} The function $\al_2 : \P(\GLS) \rightarrow \R, \; \al_2(A) = \|A\|_F^2 \si_n^{-2}(A)$ is self-convex in $\P(\GLS)$.
\end{corollary}

\begin{proof}
Note that $p : \S(\GLS) \rightarrow \P(\GLS)$ is a Riemannian submersion and $\al_2 = \al \circ p$ where $\al$ is as in Corollary \ref{cor-3b}. The corollary follows from Proposition \ref{pro-RS}.
\end{proof}

\subsection{The solution variety.}
Let us denote by $p_1$ and $p_2$ the canonical maps
$$\S_1 \stackrel{p_1}{\rightarrow} \P\left(\K^{n \times (n+1)}\right) \ \mbox{ and } \
\S_2 \stackrel{p_2}{\rightarrow} \P\left(\K^{n+1}\right) = \P_n (\K), $$
where $\S_1$ is the unit sphere in $\K^{n \times (n+1)}$ and $\S_2$ is the unit sphere in $\K^{n+1}$. Consider the affine solution variety,
$$\hat{\CW}^> = \left\{(M,\ze) \in  \S_1 \times \S_2 \ : \ M \in \mathbb{GL}_{n,n+1}^> \mbox{ and } M\ze = 0\right\}.$$
It is a Riemannian manifold equipped with the metric induced by the product metric on $\K^{n\times(n+1)}  \times \K^{n+1}$. The tangent space to $\hat\CW^>$ is given by
$$T_{(M,\ze)} \hat{\CW}^> = \left\{(\dot M, \dot \ze) \in T_M\S_1 \times T_\ze \S_2 \ : \ \dot M \ze + M \dot \ze = 0 \right\}.$$
The projective solution variety considered here is
$$\CW^> = \left\{(p_1(M),p_2(\ze)) \in  \PKn \times \Pn \ : \ M \in \mathbb{GL}_{n,n+1}^>  \mbox{ and } M\ze = 0\right\},$$
that is also a Riemannian manifold equipped with the metric induced by the product metric on $\PKn \times \Pn$.

\vskip 3mm Let us denote by $\pi_1$ the restriction to $\hat{\CW}^>$ of the first projection $\S_1 \times \S_2 \rightarrow \S_1$,
and by $R : \hat{\CW}^> \rightarrow \R$, $R = \sigma_n \circ \pi_1.$ We have

\begin{lemma}\label{lem-der1dies}
Let $w = ( M, \zeta) \in \hat\CW^>$ and let $\gamma$ be a geodesic in $\hat\CW^>$, $\gamma(0)=w$. Then,
\[
D \sigma_n(\pi_1(w))(\pi_1\circ\gamma)''(0)<0.
\]
\end{lemma}
\begin{proof} Our problem is invariant by unitary change of coordinates. Hence, using a singular value decomposition,  we can assume that $M = (\Si, 0) \in \mathbb{GL}_{n,n+1}^>$, where $\Si = \diag(\si_1\geq\cdots\geq \si_{n-1}>\si_n) \in \K^{n \times n}$ and $\ze = e_{n+1} = (0, \ldots , 0, 1)^T \in \S_2$. As $\gamma=(M(t),\zeta(t))$ is a geodesic of $\hat\CW^>\subseteq\K^{n\times(n+1)}\times\K^n$, $\gamma''(0)$ is orthogonal to $T_w\hat{\CW}$, which contains all the pairs of the form $((A,0),0)$ where $A$ is a $n\times n$ matrix, ${\rm Re}\langle \Sigma , A \rangle=0$. Hence, $M''(0)$ has the form
\[
M''(0)=(a \Si,*),
\]
for some real number $a\in\R$. Finally, $M(t)$ is contained in the sphere so $\left\| M(t) \right\|_F = 1$ and
\[
0=(||M(t)||_F^2)''(0)=2||M'(0)||_F^2+2{\rm Re}\langle M(0),M''(0)\rangle=2||M'(0)||_F^2+2a,
\]
so that $a=-\|M'(0)\|_F^2$ and $(M''(0))_{nn}=-\|M'(0)\|_F^2\sigma_n$. From Proposition \ref{pro-4},
\[
D \sigma_n(\pi_1(w))(\pi_1\circ\gamma)''(0)={\rm Re}((\pi_1\circ\gamma)''(0)_{nn})={\rm Re}(M''(0))_{nn}<0.
\]
\end{proof}

\begin{theorem} \label{th-3} The map $\al : \hat\CW^> \rightarrow \R$ given by $\al(M,\zeta) = \sigma_n(M)^{-2}$ is self-convex.
\end{theorem}

\begin{proof} Using unitary invariance we can take $M = (\Si, 0) \in \mathbb{GL}_{n,n+1}^>$, where $\Si = \diag(\si_1\geq\cdots\geq \si_{n-1}>\si_n) \in \K^{n \times n}$ and $\ze = e_{n+1} = (0, \ldots , 0, 1)^T \in \S_2$. According to proposition \ref{pro-3} we have to prove that
$$2 \left\| \dot w \right\|_w^2 \left\| DR(w) \right\|^2 \geq D^2 R^2(w)(\dot w,\dot w)$$
for every $w \in \hat\CW^>$ and $\dot w \in T_w \hat\CW^>$. From Proposition \ref{pro-4} we have
$$DR(w)\dot w = D \sigma_n(\pi_1(w))(D\pi_1(w)\dot w)={\rm Re}(D\pi_1(w)\dot w)_{nn},$$
so that $\|DR(w)\|=1$. On the other hand, assume that $\dot w\neq0$ and let $\gamma$ be a geodesic in $\hat\CW^>$, $\gamma(0)=w,\dot\gamma(0)=\dot w$. From Lemma \ref{lem-der1dies},
$$D^2R^2(w)(\dot w, \dot w) =(\sigma_n^2 \circ \pi_1\circ\gamma)''(0)=  $$
$$D^2 \sigma_n^2(\pi_1(w))(D\pi_1(w)\dot w,D\pi_1(w)\dot w)+2\sigma_nD \sigma_n(\pi_1(w))(\pi_1\circ\gamma)''(0)<$$
$$D^2 \sigma_n^2(\pi_1(w))(D\pi_1(w)(\dot w),D\pi_1(w)(\dot w)).$$
Thus, we have to prove that for $\dot y\in\K^{n\times(n+1)}$,
$$2 \left\|\dot y \right\|^2 \geq D^2\sigma_n^2(\pi_1(w))(\dot y,\dot y).$$
which is a consequence of our Proposition \ref{pro-4}. \end{proof}

\begin{corollary} \label{cor-4} The map $\al_2 : \CW^> \rightarrow \R$ given by $\al_2 (M,\zeta)=\|M\|_F^2/\si_n^2(M)$ is self-convex.
\end{corollary}

\begin{proof}
Consider the Riemannian submersion
\[
p_1\times p_2:\S_1\times \S_2\longrightarrow\PKn\times\Pn,\ p_1\times p_2(M,\ze)=(p_1(M),p_2(\ze)).
\]
Note that $T_{(M,\ze)}\hat{\CW}^>$ contains the kernel of the derivative $D(p_1\times p_2)(M,\ze)$. Thus, the restriction $p_1\times p_2:\hat{\CW}^>\rightarrow \CW^>$, is also a Riemannian submersion. The corollary follows combining Proposition \ref{pro-RS} and Theorem \ref{th-2}.
\end{proof}

\section{Self-convexity of the distance from a submanifold of $\R^j$}

Let $\CN$ be a $C^k$ submanifold without boundary $\CN \subset \R^j$, $k \ge 2$. Let us denote by
$$\rho(x) = d(x,\CN) = \inf_{y \in \CN} \left\| x-y \right\|$$
the distance from $\CN$ to $x \in \R^j$ (here $d(x,y)=\left\| x-y \right\|$ denotes the Euclidean distance). Let $\CU$ be the largest open set in $\R^j$ such that, for any $x \in \CU$, there is a unique closest point from $\CN$ to $x$.
This point is denoted by $K(x)$ so that we have a map defined by
$$K : \CU \rightarrow \CN, \ \rho(x) = d(x,K(x)).$$
Classical properties of $\rho$ and $K$ are given in the following (see also Foote \cite{foo}, Li and Nirenberg \cite{li-nir}).

\begin{proposition} \label{pro-5} \begin{enumerate} \item $\rho$ is defined and $1-$Lipschitz on $\R^j$,

\item For any $x \in \CU$, $x-K(x)$ is a vector normal to $\CN$ at $K(x)$ i.e. $x-K(x) \in \left(T_{K(x)}\CN\right)^\perp$,
\item $K$ is $C^{k-1}$ on $\CU$,
\item $\rho^2$ is $C^{k}$ on $\CU$, $D\rho^2(x)\dot x =  2 \left\langle x - K(x), \dot x \right\rangle$ and $D^2\rho^2(x)(\dot x,\dot x)=2\|\x\|^2  - 2 \left\langle DK(x)\dot x , \dot x \right\rangle$
\item $\rho$ is $C^{k}$ on $\CU \setminus \CN$,
\item $\left\langle DK(x)\dot x , \dot x \right\rangle \ge 0$ for every $x \in \CU$ and $\dot x \in \R^j$.
\end{enumerate}
\end{proposition}

\proof \begin{enumerate} \item For any $x$ and $y$ one has $\rho(x) = d(x,K(x)) \le d(x,K(y)) \le d(x,y) + d(y,K(y)) = d(x,y) + \rho(y)$. Since $x$ and $y$ play a symmetric role we get $\left|\rho(x)-\rho(y)\right| \le d(x,y).$

\item This is the classical first order optimality condition in optimization.
\item This classical result may be derived from the inverse function theorem applied to the canonical map defined on the normal bundle to $\CN$
$$\mbox{can} : \mathrm{N}\CN \rightarrow \R^j, \ \mbox{can}(y,n)=y+n,$$
for every $y \in \CN$ and $n \in N_y\CN = \left(T_{y}\CN\right)^\perp$. The normal bundle is a $C^{k-1}$ manifold, the canonical map is a $C^{k-1}$ diffeomorphism when restricted to the set $\{(y,n):y+tn\in\CU,\ \forall\ 0\leq t\leq 1\}$ and
$K(x)$ is easily given from $\mbox{can}^{-1}$.
\item The derivative of $\rho^2$ is equal to
$D\rho^2(x)\dot x = 2 \left\langle x - K(x), \dot x - DK(x)\dot x \right\rangle = 2 \left\langle x - K(x), \dot x \right\rangle$ because $DK(x)\dot x \in T_{K(x)} \CN$ and $x-K(x) \in \left(T_{K(x)}\CN\right)^\perp$. Thus $\nabla \rho^2(x) = 2(x-K(x))$ is $C^{k-1}$ on $\CU$ so that $\rho^2$ is $C^{k}$. The formula for $D^2\rho^2$ follows.
\item Obvious.
\item Let $x(t)$ be a curve in $\CU$ with $x(0) = x$. Let us denote $\frac{dx(t)}{dt} = \dot x(t)$, $\frac{d^2x(t)}{dt^2} = \ddot x(t)$, $y(t) = K(x(t))$,
$\frac{dy(t)}{dt} = \dot y(t)$ and $\frac{d^2y(t)}{dt^2} = \ddot y(t)$. From the first order optimality condition we get
$$\left\langle x(t) - y(t), \dot y(t) \right\rangle = 0$$
whose derivative at $t=0$ is
$$\left\langle \dot x - \dot y, \dot y \right\rangle + \left\langle x - y, \ddot y \right\rangle = 0.$$
Thus
$$\left\langle DK(x)\dot x , \dot x \right\rangle = \left\langle \dot y, \dot x \right\rangle = \left\langle \dot y, \dot y \right\rangle - \left\langle x - y, \ddot y \right\rangle.$$
This last quantity is equal to $\left.\frac{1}{2}\frac{d^2}{dt^2}\left\|x-y(t)\right\|^2\right|_{t=0}$. It is nonnegative by the second order optimality condition.
\end{enumerate}
\qed

\vskip 3mm \noindent {\bf Proof of Theorem \ref{th-2} and Corollary \ref{cor-th-2}.} We are now able to prove our second main theorem. Let us denote $\al(x) = 1/{\rho(x)^2}.$ We shall prove that $\al$ is self-convex on $\CU$.
From proposition \ref{pro-3} it suffices to prove that, for every $\dot x \in \R^j$,
\[
2\|\x\|^2\|D\rho(x)\|^2\geq D^2\rho^2(x)(\x,\x)
\]
or, according to Proposition \ref{pro-5}.4 and $\left\| D\rho \right\| = 1$, that
\[
2\|\x\|^2 \geq 2\|\x\|^2  - 2 \left\langle DK(x)\dot x , \dot x \right\rangle.
\]
This is obvious from Proposition \ref{pro-5}.4.

Now we prove Corollary \ref{cor-th-2}. Let $\S_1(\R^j)$ be the sphere of radius $1$ in $\R^j$ and let $p_{\R^j}$ denote the canonical projection $p_{\R^j} : \R^j \rightarrow \P(\R^j)$. Note that the preimage of $\CN$ by $p_{\R^j}$ satisfies
$$d(y, p_{\R^j}^{-1}(\CN)) = d_\P (p_{\R^j}(y), \CN)  \| y \|.$$
As in the proof of Corollary \ref{cor-3b},

the mapping $1/{\rho(x)^2}$ is self-convex in the set $\S_1(\R^j) \cap p_{\R^j}^{-1}(\CU)$. Now, apply Proposition \ref{pro-RS} to the Riemannian submersion $p_{\R^j}$ to conclude the corollary.
\qed

\vskip 3mm \noindent {\bf Two examples.}
\begin{example} \label{ex-circle} Take $\CU$ the unit disk in $\R^2$ and $\CN$ the unit circle.
The corresponding function is given by
$$\al(x) = d(x, \CN)^{-2} = 1 / \left(1 - \left\| x \right\| \right)^2.$$
According to Theorem \ref{th-2}, the map $\log \al (x)$ is convex along the condition geodesics in
$$ \CU \setminus \left\{\left(0,0\right)\right\} = \left\{x \in \R^2 \ : \ 0 < \left\| x \right\| < 1 \right\}.$$
This property also holds in $\CU$: a geodesic through the origin is a ray $x(t) = (-1 + e^t)(\cos \theta, \sin \theta)$ when
$-\infty < t \leq 0$, and $x(t) = (1 - e^{-t})(\cos \theta, \sin \theta)$ when $0 \le t < \infty$ for some $\theta$. In that case
$$\log \al (x(t)) =  2 \left| t \right|$$
which is convex.
\end{example}

\begin{example} \label{ex-two-points} Take $\CN \subset \R^2$ equal to the union of the two points $(-1,0)$ and $(1,0)$.
In that case
$$\al(x)^{-1} = d(x,\CN)^{2} = \min \left((1+x_1)^2 + x_2^2, (1-x_1)^2 + x_2^2 \right).$$
It may be shown that for any $0 < a \leq 1/10$, the straight line segment is the only minimizing geodesic joining
the points $(0,-a)$ and $(0,a)$. Since $\log \al(0, t) = - \log(1+t^2)$
has a maximum at $t = 0$, $g(t)$, $-a \leq t \leq a$, cannot be log-convex. Here $\left\{ 0 \right\} \times \R$ is equal to the locus in $\R^2$ of points equally distant from the two nodes which is the set we avoid in Theorem \ref{th-2}.
\end{example}


text:
\begin{document}

\title{ Answer To Graham Higman's Question Regarding Januarials}
\author{Qaiser Mushtaq and Saadia Mehwish }

\begin{center}
{\LARGE Januarials of simple and general type}

Saadia Mehwish$^{1}$ and Qaiser Mushtaq$^{2}$
\end{center}

$^{1}${\footnotesize Department\ of\ Mathematics,}${\footnotesize \ }$
{\footnotesize The Islamia University of Bahawalpur, Bahawalpur 63100,
Pakistan.}

\ {\footnotesize E-mail: saadiamehwish@gmail.com.}

$^{2}${\footnotesize The Islamia University of Bahawalpur, Bahawalpur 63100,
Pakistan.}

\ {\footnotesize E-mail: pir\_qmushtaq@yahoo.com.}\

\begin{center}
\bigskip \textbf{Abstract}
\end{center}

{\small Januarials were defined by Graham Higman in his last series of
lectures. In this paper we answer some questions posed by Higman in these
lectures.}

\begin{center}
{\small \bigskip }
\end{center}

{\tiny AMS Classification[2010]: Primary 05C25; Secondary 20G40}

{\tiny Keywords: Coset diagrams, Januarials, Triangle groups, Hecke groups
and genus.}

{\tiny \bigskip }

\section{Introduction}

\bigskip Coset diagrams were introduced by Graham Higman as a diagram
representing an action of a particular group on a set or a space.

Several papers, namely \cite{br1, br, qmf, ww, at} appeared in which these
coset diagrams were used to prove interesting topological and group
theoretical results. Q. Mushtaq \cite{Qm} laid the foundations of coset
diagrams for the action of the modular group on finite Galois fields and
real quadratic irrational number fields.

Graham Higman introduced the concept of januarials in his lectures at Oxford
in 2000, which turned out to be his last -- see \cite{jan}. In these he
posed several questions. This paper answers two of them -- see Proposition
\ref{p} and Theorem \ref{t} below.

\section{Januarials}

\label{januarials}

Let $\Delta=\Delta(2,k,\ell)$ be the abstract group with presentation
\begin{equation*}
\Delta (2,k,\ell )=\left\langle x,y:x^{2}=y^{k}=(xy)^{\ell}=1\right\rangle
\end{equation*}
where $k,\ell\in\ams{Z}^{\geq 2}\cup\infty$. Such groups are called triangle
groups as they act discretely on the $2$-sphere, Euclidean plane or
hyperbolic plane with fundamental region a triangle. The $\Delta(2,k,\infty)$
are called \emph{Hecke groups\/}; in particular when $k=3$ we obtain the
classical modular group.

Suppose that we have an action of $\Delta $ on some set $S$. A \emph{coset
graph\/} $\Gamma $ is a directed graph in the plane depicting the action as
follows: the vertices of the graph are the elements of the set $S$; if two
vertices are transposed by the action of $x$ then there is an undirected $x$
-edge connecting them; vertices fixed by $x$ have no $x$-edge incident with
them; there is a directed $y$-edge from $u$ to $v$ when $u$ is sent to $v$
under the action of $y$.

Now associate to each vertex of the coset graph a permutation of the
vertices incident to it as in Figure \ref{figure:123}.

\begin{figure}
  \centering
\BoxedEPSF{123a.eps scaled 750}
  \caption{$2$-cell
embedding using the method described by \protect\cite[Theorem
6.47]{white84}}
\label{figure:123}
\end{figure}

Then by \cite[Theorem 6.47]{white84} we have a uniquely defined $2$-cell
embedding of $\Gamma $ into an orientable surface where the $2$-cells have
boundary labels $y^{n}$, with $n$ dividing $k$, or $(xy)^{m}$, with $m$
dividing $\ell $. Call such $2$-cells \emph{$y$-faces\/} or \emph{$xy$
-faces\/} repectively.

The result is called a \emph{coset diagram\/}. The picture above shows a $
\Gamma$ depicting the action of $\Delta(2,3,3)$ on a set of size six and the
resulting diagram embedded in the $2$-sphere.

From now we can omit the orientations on the $y$-edges of a coset diagram
with the understanding that the boundary edges of each $y$-face are oriented
anticlockwise around the face.

Suppose we have an action of some $\Delta $ on a set $S$ such that there are
two orbits of $\langle xy\rangle $, each consisting of $|S|/2$ elements. A
\emph{januarial\/} $J$ is the resulting coset diagram. In particular, a
januarial is a diagram with two $xy$-faces. The examples in Figure 1 and 2\
are thus januarials. For a nice introduction to januarials, see \cite{jan},
from which the following concepts are taken.

Let $S_1$ and $S_2$ be the discs that result from taking the closures of the
two $xy$-faces.

It is convenient to collapse superfluous structure in a graph or diagram.
Let $J$ be a januarial arising from the coset graph $\Gamma $. Collapsing
all the $y$-faces to a point gives the \emph{companion graph\/} $\Gamma
^{\prime }$ and the \emph{companion diagram\/} $J^{\prime }$. Let $
S_{1}^{\prime },S_{2}^{\prime }$ be the result of applying this to the discs
$S_{1},S_{2}$.

\begin{figure}[h]
  \centering
\BoxedEPSF{234.eps scaled 750}
  \caption{A graph of a
januarial $J$, its \emph{companion graph\/} $\Gamma ^{\prime }$ and the
\emph{common graph\/} $\Upsilon$.}
\end{figure}

Let $g$ be the genus of the surface in which the januarial $J$ is embedded.
Let $R_{i}$ be a small closed neighbourhood of $S_{i}^{\prime }$ in $
J^{\prime }$. Then $R_{i}$ is a closed surface with boundary. Let $g_{i}$ be
its genus and $h_{i}$ is the number of connected components in the boundary.

The \emph{common graph\/} $\Upsilon $ is the intersection $S_{1}^{\prime
}\cap S_{2}^{\prime }$. The following is in \cite[Lemma 3.5]{jan}:

\begin{lemma}
Let $P_{1}$ (resp. $P_{2}$) be the set of all paths that traverse successive
edges in $\Upsilon $ in the directions they are traversed by $S^{\prime
}_{1} $ (resp. $S^{\prime }_{2}$) in such a way that whenever such a path
reaches a vertex, it continues along the right-most of the other edges in $
\Upsilon$ incident with that vertex. (The next edge is necessarily traversed
by $S_{1}$ (resp. $S_{2}$) in that direction.) All such paths close up into
circuits, and $P_{1}$ (resp. $P_{2}$) partitions $\Upsilon$, in the sense
that the union of the circuits is $\Upsilon $ and no two share an edge. The
cardinality of $P_{1}$ (resp. $P_{2}$) is $h_{2}$ (resp. $h_{1}$).
\end{lemma}

A januarial is of \emph{simple type\/} if $\Upsilon$ is composed of $h$
disjoint simple circuits. In this case $h=h_1=h_2$ and we denote it by $
(h,g_1,g_2)$. Otherwise, the januarial is of \emph{general type\/} $
((h_{1},g_{1}),(h_{2},g_{2}))$.

The following are Lemmas 3.1, 3.4 and 3.6 of \cite{jan}:

\begin{lemma}
\label{1}Twice the genus $g$\ of a januarial equals the number $E$ of $x$
-edges which are not loops minus the number $V$ of $y$-faces.
\end{lemma}

\begin{lemma}
\label{2}The genus of a januarial $J$ of simple type is $g=g_{1}+g_{2}+h-1$.
\end{lemma}

\begin{lemma}
\label{3}The $g_{i}$ satisfy $2-2g_{i}=V_{i}-E_{i}+h_{i}+1$ where $V_{i}$
and $E_{i}$ denote the number of vertices and edges in the subgraph of $
\Gamma ^{\prime }$ visited by $S_{i}^{\prime }.$
\end{lemma}

In this paper we first generalize the formula in lemma \ref{2} for obtaining
the genus of a januarial of simple type to that of general type. Then we
address the questions posed by Graham Higman.

\begin{problem}
For a given $k$, what are the possible values for and interrelationships
between $h,g_{1}$ and $g_{2}$ for januarials of simple type? And a similar
question for januarial of general type.

Are there arbitrarily large values of $k$ for which there exist examples
with $h=1$?
\end{problem}

In \cite{qmsm} we describe a method that constructs a januarial from an
action of $\Delta _{k}$ on the projective line $PL(F_{p})$ over the field of
order $p$, with the property that $xy$ acts as permutation of $PL(F_{p})$ of
finite order $\ell$. Necessarily $2\ell =p+1$ for a januarial. It turns out
that there is a polynomial
\begin{equation*}
f_{\ell }= \left\{
\begin{array}{c}
{\displaystyle \binom{\ell -1}{0}\theta ^{(\ell -1)/2}-\binom{\ell -2}{1}
\theta ^{(\ell -3)/2}+\binom{\ell -3}{2}\theta ^{(\ell -5)/2}-...\text{ for
odd }\ell} \\
\\
{\displaystyle \binom{\ell -1}{0}\theta ^{\ell /2-1}-\binom{\ell -2}{1}
\theta ^{\ell /2-3}+\binom{\ell -3}{2}\theta ^{\ell /2-5}-...\text{ for even
}\ell}
\end{array}
\right.
\end{equation*}
certain roots of which give linear fractional transformations of $PL(F_p)$.
For, let $\theta $ be a root of $f_{\ell }$ that is not a root of $f_{\ell
/s}$ where $s\mid \ell $. Let $X$, $Y$ be the linear fractional
transformations of $PL(F_{p})$ given by
\begin{equation*}
X:z\mapsto \frac{az+cd}{cz-a} \text{ and } Y:z\mapsto \frac{ez+fd}{fz+b-e}
\end{equation*}

where $a,b,c,d,e,f\in

\mathbb{Z}

$, with $\nabla =-(a^{2}+dc^{2})\neq 0$, $r=a(2e-b)+2dcf$ and $
1+df^{2}+e^{2}-eb=0$ and $\theta \nabla =r^{2}$.

Then the action of $\Delta _{k}$\ on the $p+1$ points of $PL(F_{p})$ gives a
januarial. Write $\bar{x}$ and $\bar{y}$ for the permutations induced by $X$
and $Y$. We say that this januarial has been constructed from the Hecke
group $\Delta_k$.

In \cite{jan} a technique is proposed for finding januarials. Associates are
constructed of known coset diagrams and investigated. Q. Mushtaq and S.
Mehwish \cite{qmsm} used the method of parametrization described in \cite
{Qm, poly} to construct januarials from Hecke groups. In this paper we
construct januarials for the natural action of some permutation groups that
are homomorphic images of triangle groups.

\section{\label{6}Relationship between $h_{1},h_{2},g_{1}$ and $g_{2}$}

We now obtain the generalized formula for the genus of a januarial of
general type:

\begin{lemma}
\label{4}Let $\alpha =v-e$ where $v$ is the number of vertices in $\Upsilon $
and $e$ is the number of edges. Then the genus of a general type januarial $
((h_{1},g_{1}),(h_{2},g_{2}))$ is
\begin{equation*}
g_{1}+g_{2}+(h_{1}+h_{2}+\alpha )/2-1.
\end{equation*}
\end{lemma}

\begin{proof}
By Lemma \ref{1} the genus $g$ of a januarial is $\ g=\frac{1}{2}(E-V)$ and
by Lemma \ref{3} $g_{i}=\frac{1}{2}(E_{i}-V_{i}-h_{i}+1)$. Since $
E=E_{1}+E_{2}-e$ and $V=V_{1}+V_{2}-v,$ we get $g=\frac{1}{2}(E_{1}+E_{2}-e$
$-(V_{1}+V_{2}-v)).$ Simple calculations using the above lead to the
required result $g=g_{1}+g_{2}+\frac{1}{2}(h_{1}+h_{2}+\alpha )-1.$
\end{proof}

\vspace*{1em}

In \cite{qmsm} it is proved that the genus $g$ of januarials constructed
from Hecke Groups is given by
\begin{equation*}
g=-\frac{p+1-\eta _{y}}{2k}+\frac{1}{4}(p+1-2\eta _{y}-\eta _{x}),
\end{equation*}
where $\eta _{x}$ and $\eta _{y}$ are the number of fixed points of $x$ and $
y$ respectively. This equation implies that $k$ and $p$ are proportional.
Moreover, the number of fixed points of $x$ and $y$ are the same for fixed
values of $k$ and the prime $p$. Thus:

\begin{lemma}
\label{5} The genus of januarials constructed from Hecke groups for fixed
values of $k$ and the prime $p$, is a fixed value $g_{pk}$.
\end{lemma}

From lemmas \ref{2}, \ref{4} and \ref{5} we have:

\begin{proposition}
\label{p}When a januarial is constructed from a Hecke Group with $k$ and the
prime $p$ fixed, we have

\begin{enumerate}
\item $g_{1}+g_{2}+h=g_{pk}+1$ for januarials of simple type and,

\item $g_{1}+g_{2}+(h_{1}+h_{2}+\alpha )/2=g_{pk}+1$ for januarials of
general type.
\end{enumerate}
\end{proposition}

That is, for any $p$ and $k$ the sums $g_{1}+g_{2}+h$ and $
g_{1}+g_{2}+(h_{1}+h_{2}+\alpha )/2$ are constant in these two cases.

\section{\label{7}Januarials with a simple disjoint circuit}

Analyzing januarials constructed from Hecke groups as described at the end
of Section \ref{januarials} it is observed that for higher values of $k$ the
chances of getting a januarial with simple disjoint circuits decreases.
Moreover the value of $p$ affects the number of circuits in the januarial.
Therefore finding a simple januarial with $h=1$ from a Hecke group becomes
almost impossible as we move to higher values of $k$. These remarks are made
precise in this section.

\begin{proposition}
\label{pro}The valency of each vertex of the subgraph $\Upsilon $ of a $k$
-januarial is even.
\end{proposition}

\begin{proof}
Each vertex in the subgraph $\Upsilon $ is a $y$-face of the $k$-januarial
and the edges are $x$-monogons. Let $E$ be an $x$-monogon between the
vertices $U$ and $V$ of $\Upsilon .$

Since $\Upsilon =S_{1}^{\prime }\cap S_{2}^{\prime },$ the $x$-monogon $E$
lies both in the restriction of $S_{1}^{\prime }$ to $\Upsilon $ and in the
restriction of $S_{2}^{\prime }$ to $\Upsilon .$ If in the restriction of $
S_{1}^{\prime }$ to $\Upsilon$ the monogon $E$ takes $U$ to $V$ then in the
restriction of $S_{2}^{\prime }$ to $\Upsilon$ we have that $E$ takes $V$ to
$U.$ In other words $E$ is unidirectional in both the restrictions.

Now for the vertex $U$ in the restriction of $S_{1}^{\prime }$ to $\Upsilon$
the monogon $E$ is an outgoing edge for some circuit in $P_{1}.$ There must
then be a unidirectional incoming edge $E^{\prime }$ at $U$ in that circuit.

This implies that at any vertex of $\Upsilon$, incident edges exist
pairwise. So the valency of each vertex of the subgraph $\Upsilon $ of a $k$
-januarial is even.
\end{proof}

\begin{theorem}
Every $3$-januarial is of simple type.
\end{theorem}

\begin{proof}
By Proposition \ref{pro}, at any vertex of the subgraph $\Upsilon $ of a $3$
-januarial only two edges can be incident while moving along the orbit: one
pointing towards the vertex and the other pointing away from the vertex. So
only disjoint simple circuits will be formed in the subgraph $\Upsilon$,
giving it a simple $3$-januarial.
\end{proof}

\begin{figure}
  \centering
\BoxedEPSF{D17_17_8.eps scaled 500}
  \caption{A diagram for $\Delta(17,17,8)$. Note that there is an
    $x$-edge connecting the vertices $4$ and $13$.}
\label{figure:D(17,17,8)}
\end{figure}

\vspace*{1em}

When $k\geq 4$ the likelihood of getting simple $k$-januarials decreases
expeditiously. For an example we construct an $8$-januarial from the Hecke
group $\Delta _{8}$ acting on $PL(F_{17})$. We have $\ell =9$ and hence
\begin{equation*}
f_{9}=1\theta ^{4}-8\theta ^{3}+21\theta ^{2}-20\theta +5.
\end{equation*}
The roots of $f_{9}$ yielding januarials are $9,15$ and $16$. For an $8$
-januarial, if we consider the root $\theta =9$ we get the linear fractional
transformations
\begin{equation*}
X:z\mapsto \frac{z+10}{10z-1}\text{ and }Y:z\mapsto \frac{4}{4z+8}
\end{equation*}

The action of $X$ and $Y$ on $PL(F_{17})$ gives
\begin{eqnarray*}
\bar{x} &=&(0,7)(1,5)(2,6)(3,11)(4,13)(8,14)(9)(10,16)(12,\infty )(15) \\
\bar{y} &=&(0,9,14,16,1,6,15,\infty )(2,13,8,12,11,4,3,7)(5)(10) \\
\bar{x}\bar{y} &=&(0,4,7,19,13,1,17,5,22,11,14,18)(\infty
,20,6,16,8,3,9,15,10,2,12,21).
\end{eqnarray*}
Hence the coset diagram $D(17,17,8)$ is given by Figure
\ref{figure:D(17,17,8)}. The two partitions of $\Upsilon $ into
circuits is drawn in Figure \ref{figure:D(17,17,8)_common}.

\begin{figure}
  \centering
\BoxedEPSF{D17_17_8_common.eps scaled 750}
  \caption{}
\label{figure:D(17,17,8)_common}
\end{figure}

This is an 8-januarial of general type with $h_{1}=2$ and $h_{2}=1.$ It has $
8$ edges which are not loops and $4$ $y$-faces, giving genus $g=\frac{1}{2}
(8-4)=2.$ Calculating the genera using $2-2g_{i}=V_{i}-E_{i}+h_{i}+1\,$\
gives $g_{1}=\frac{1}{2}(1-2+5-2)=1$ and $g_{2}=\frac{1}{2}(1-4+6-1)=1.$
Alternatively by Lemma 5 the genus of this general type januarial $
((2,1),(1,1))$ is $g=1+1+(2+1-1)/2-1=2.$

\begin{figure}
  \centering
\BoxedEPSF{even_simple.eps scaled 750}
  \caption{}
\label{figure:even_simple}
\end{figure}

From the above example it follows that the action of Hecke groups $\Delta_k$
for larger $k$'s on smaller prime fields do not yield simple januarials.
This leads to the question of whether there exist examples of simple
januarials for larger values of $k$. If so, then this answers Higman's
question of whether there are arbitrarily large values of $k$ for which
there exist examples with $h=1$:

\begin{theorem}
\label{t}There exist simple januarials with $h=1$ for all values of $k$.
\end{theorem}

By taking into consideration the definition of januarials and our
requirement that we have one simple disjoint circuit in the common graph, in
the proof below we construct the $k$- januarials for even and odd $k$
separately.

\begin{proof}
Consider the diagram in Figure \ref{figure:even_simple}. It is easy to see it as a permutation action of
the subgroup generated by
\begin{equation*}
\bar{x}=\left( 1,3k/2+1\right) \left( k/2+1,k+1\right) \text{ and }\bar{y}
=\left( 1,2,\ldots ,k\right) \left( k+1,k+2,\ldots ,2k\right)
\end{equation*}
of $S_{2k}.$

This is a $k$-januarial for even values of $k$. The companion graph and the
two partitions of the subgraph $\Upsilon$ are show in Figure \ref{figure:even_simple_common}.

\begin{figure}[h]
  \centering
\BoxedEPSF{even_simple_common.eps scaled 750}
  \caption{}
\label{figure:even_simple_common}
\end{figure}

This shows that the januarial is of simple type with $h=1$. For odd values of $k$
consider the diagram in Figure \ref{figure:odd_simple} which shows a permutation action of $\Delta
(2,k,2k)$ on 4k points.

\begin{figure}[h]
  \centering
\BoxedEPSF{odd_simple.eps scaled 750}
  \caption{}
\label{figure:odd_simple}
\end{figure}

It is a $k$-januarial and the two faces of $xy$ are labeled by $1$ $2$ $
3\ldots \ 2k$ and $2k+1$ $2k+2$ $2k+3\ldots \ 4k$. The companion graph and
the two partitions of the subgraph $\Upsilon $ are shown in Figure \ref{figure:odd_simple_common}.

\begin{figure}[h]
  \centering
\BoxedEPSF{odd_simple_common.eps scaled 750}
  \caption{}
\label{figure:odd_simple_common}
\end{figure}

This shows that the januarial is of simple type with $h=1$.
\end{proof}

\vspace*{1em}

These januarials are the result of actions of $\Delta (2,k,\ell )$ on the
set of $2k$ and $4k$ points for even and odd values of $k$ respectively,
which can easily be extended to larger finite sets in many ways. For
example, we can insert $k$-polygons in the diagrams above while maintaining
the symmetry. This technique is also used by Q. Mushtaq and F. Shaheen in
\cite{qmf}. Thus, there exist many more januarials with the required
condition.

config: arxiv


TITLE: Let $R$ be an integral domain. If $I \cap J = IJ$ for all ideals $I,J$ of $R,$ then $R$ is a field.
QUESTION [0 upvotes]: Let $R$ be an integral domain. If $I \cap J = IJ$ for all ideals $I,J$ of $R,$ how do I show $R$ is a field? Hints will suffice. Thank you.

REPLY [1 votes]: Arithmetically: since ideal intersection $= \rm lcm$ for principal ideals, the hypothesis implies for all $\,a,b\neq 0\,$ we have $\,{\rm lcm}(a,b) = ab,\,$ so $\,\gcd(a,b)=1,\,$ i.e. $\,a,b$ have only unit common factors. Hence for $\,b=a\,$ the common factor $\,a\,$ of $\,a,b$ is a unit, so $\,a\neq 0\,\Rightarrow\, a\,$ is a unit, so $R$ is a field.


TITLE: Energy-momentum relation in SR and GR
QUESTION [5 upvotes]: There is the famous energy-momentum relation
$$E^2 - p^2 c^2 = m^2 c^4.$$
I thought it is always valid. "Always" means it is valid in general, so in GR and SR. It is something like an elemantary construction. But now I am not sure and if I understand right it is valid only in SR. Right?
Below I write some more calculations and thoughts for a better understanding of my question.

In SR it is very simple to derive the formula. We start with the metric
$g_{\mu\nu} = \eta_{\mu\nu} = {\rm diag}(1,-1,-1,-1)$
and the product
$g_{\mu\nu} p^\mu p^\nu = m^2 c^2$
where $p^\mu = m u^\mu = m \frac{{\rm d} x^\mu}{{\rm d} \tau}$ is the four-momentum. This gives
$g_{\mu\nu} p^\mu p^\nu = (p^0)^2 + \eta_{ij} p^i p^j = (E/c)^2 - p^2 = m^2 c^2$
with the definitions $p^0 = E/c$ and $\eta_{ij} p^i p^j = {\mathbf{p}}^2 = p^2$. The last equation is equivialent to the above energy-momentum relation.

Now, when we turn to GR and consider the general metric in spherical coordinates
$g_{\mu\nu} = {\rm diag}(g_{00}, g_{11}, -r^2, -r^2 \sin^2 \vartheta)$
gives
$g_{\mu\nu} p^\mu p^\nu = g_{00} (p^0)^2 + g_{ij} p^i p^j = g_{00} (p^0)^2 - p^2 = m^2 c^4$
Here we have to be carefully now with the definition of $p^0$. In nearly all books about GR we find for the above spherical metric the following relation (or equivalent)
$g_{00} m \frac{{\rm d} x^\mu}{{\rm d} \tau} = E/c = \rm const$
which is interpreted as the conservation of energy. In SR we have $g_{00} = \eta_{00} = 1$ and therefore $p^0 = E/c$. But in GR we have the $g_{00}$ coefficent, e.g.
$g_{00} p^0 = E/c$
This would give the following energy-momentum relation
$\frac{E^2}{g_{00}} - p^2 c^2 = m^2 c^4$
Am I missing or do I mix something, e.g the definition of the time-component $p^0$ through an energy?
Or are the calcutions all right and in GR the energy-momentum relation is indeed different compared to the above famous and well known, which seems to be valid only in SR.
I am heavily confused.

My only idea how to resolve this is to redefine the energy in the general case, e.g.
$\sqrt{g_{00}} p^0 = \mathcal{E}/c$
Note the square root compared to the above definition. This would give an epxression analogue to the famous relation
$\mathcal{E}^2 - p^2 c^2 = m^2 c^4$
But $\mathcal{E}$ is not conserved here any more, $\mathcal{E} \neq \rm const$. Instead we have $\sqrt{g_{00}} \mathcal{E} = E = \rm const$.
Is this a problem?
In SR we have simply $E = \mathcal{E}$ and there is no need to worry. So, SR automatically implies that $p^0$ is conserved. But it is not necessary the case in GR, right? Or is this all just a question of definition/convention?

REPLY [1 votes]: GR is locally the same as SR -- that's one way of stating the equivalence principle. So stuff like this is exactly the same in GR as in SR. There is actually no GR involved in your question. We can use arbitrary coordinate systems in SR. GR happens when there is curvature, which is not relevant here because we're talking about a single vector, which lives in its own neighborhood of spacetime.
A couple of aspects of your presentation are needlessly complicating things. Do yourself a favor and start working in natural units, with $c=1$, so you don't have to keep writing fatcors of $c$ all over the place. Also, there is no special reason to use spherical coordinates here. GR works with any coordinate system. If you want to reduce this to its essence, just do a 1+1-dimensional metric in a coordinate system where the metric is $\operatorname{diag}(g_{00},g_{11})$.
The basic issue here is that we don't just define energy as the 0th component of the energy-momentum vector. As a trivial example of why we wouldn't do this, consider the coordinate system $(x,t)$, where $x$ and $t$ are local Minkowski coordinates, i.e., we just swap the order. In these coordinates, energy would be the 1th component, not the 0th.
In general, energy is the timelike component of the energy-momentum. For an observer with normalized velocity vector $u$, the energy of an object is $u^\mu p_\mu$.
The identity $E^2-p^2=m^2$ is always valid. It equates the norm of a four-vector to a scalar, and equations of that form are valid in all coordinate systems.


TITLE: Maximum Likelihood Estimator for a Poisson random variable
QUESTION [0 upvotes]: $X$ is distributed as a Poisson random variable with parameter $\lambda$ if it has the probability mass function:
$f(x; \lambda) = \dfrac{e^{-\lambda}\lambda^x}{x!},\ x = 0,1,2,\ldots$

Find the maximum-likelihood estimator for the parameter $\lambda$ based on a random sample of size $n$.

I understand the theory behind MLEs, but am not sure how to go about solving this problem. I did originally take the natural log, but wasn't sure what to do afterwards. I'm very new to this.

REPLY [2 votes]: The probability of observing $(x_1,\ldots,x_n)$ is $$L=f(x_1;\lambda) \ldots f(x_n; \lambda) = \frac{e^{-n \lambda} \lambda^{\sum\limits_i x_i} }{x_1! \ldots x_n!}.$$ Taking the log, you get
$$\ell = -n \lambda + \left(\sum\limits_i x_i\right) \log \lambda + C,$$ where $C$ is terms which do not depend on $\lambda$. Now, maximize $\ell$ with respect to $\lambda$ by differentiating it with respect to $\lambda$, setting it equal to zero and checking with the first or second derivative test that you do indeed have a maximizer (alternatively note that $\ell$ is concave in $\lambda$ so it is indeed the maximizer).


TITLE: Using Mathematical Induction to prove Reversal of Strings
QUESTION [0 upvotes]: The problem at hand:
Recall our recursive definition of the reversal of a string over $\Sigma$.

Base case: $\epsilon^R=\epsilon$
If $w \in \Sigma^*$ and $c \in \Sigma$, then $(wc)^R =cw^R$.

Use the definition above along with mathematical induction to prove the following: If $v,w$ are strings over $\Sigma$, then $(vw)^R= w^Rv^R$.
I'm honestly stumped and don't know where to begin. I've done mathematical induction on actual numbers but not on just variables like this problem. Any help is appreciated.

REPLY [0 votes]: You can prove it by induction on the structure of $w.$ The idea is to show that

The equation holds for $w=\epsilon$.
If the equation holds for $w'$ and $c$ is a character, then it holds for $w'c.$

Hopefully you can see how this implies it holds for any string $w$ by analogous reasoning to induction on $\mathbb N$.
(Alternatively as Lord Shark suggests in the comments, you can just think of it as ordinary induction on the length of $w.$ If $w=w'c,$ then $w'$ is a string shorter than $w.$)
The base case is easy. Then for the induction step we have $$(v w'c)^R = c(vw')^R=cw'^Rv^R=(w'c)^Rv^R$$ where the second equality uses the induction hypothesis and the other two use the definition of $R.$