text:
\section{Introduction} \label{sect:intro}
Finding an element of a subset $\mathcal{V}$ in ${\mathbb{R}}^n$ closest to a specified point $y$ is a common task in computational mathematics, often called the {\em Euclidean distance (ED) minimization problem}:
\begin{align} \label{prob:ED} \textup{minimize } \sum_{i=1}^n (x_i-y_i)^2 \,\,\textup{ subject to } \,\,x \in {\mathcal V}. \end{align}
 Our current work is motivated by the systematic study of the ``critical points'' of the problem \eqref{prob:ED} in an algebraic setting, initiated in \cite{DHOST} and continued in \cite{draisma,hwangrae,genericEDdeg}. There, the basic assumption is that
$\mathcal{V}$ is a {\em real variety} ---  zero set of finitely many polynomials with real coefficients.
Consider now the set ${\mathcal V}_{\mathbb C}$ of complex points satisfying the defining equations of $\mathcal{V}$.
Then a {\em critical point} of $y \in {\mathbb C}^n$ with respect to \eqref{prob:ED} is any smooth (possibly complex) point
$x$ of ${\mathcal V}_{\mathbb C}$ such that $y-x$ lies in the normal space of ${\mathcal V}_{\mathbb C}$ at $x$, meaning that $y-x$ lies in the span of the gradients of the defining equations at $x$. It was shown in \cite{DHOST} that for a general data point $y\in {\mathbb{C}}^n$, the number of (complex) critical points of
\eqref{prob:ED} is a constant. This constant is called the {\em Euclidean distance degree} of ${\mathcal V}$, denoted by EDdegree$({\mathcal V})$, and is a measure of the algebraic complexity of expressing a minimizer of \eqref{prob:ED} as a function of $y$.

The work in this paper is geared towards understanding the (real) critical points
of {\em orthogonally invariant} matrix sets.
Our theme is best illustrated with an example.
	Fix positive integers $r \leq n \leq t$ and consider the  matrix set
	$$
	\ensuremath{\mathbb{R}}^{n\times t}_r := \{X\in \ensuremath{\mathbb{R}}^{n\times t} \ : \  \textup{rank}(X) \leq r\}.
	$$
Finding the closest matrix of rank at most $r$ to a given matrix $Y$ arises in many applications.  The set $\ensuremath{\mathbb{R}}^{n \times t}_r$ is a real variety, and the authors of
\cite{DHOST} established that $\textup{EDdegree}(\ensuremath{\mathbb{R}}^{n\times t}_r) = {n \choose r}$. They also provide a recipe for
all the critical points of $Y$ on $\ensuremath{\mathbb{R}}^{n\times t}_r$, which may be viewed as a generalization of the {\em Eckart-Young theorem}.

In the context of this paper, what is important about $\ensuremath{\mathbb{R}}^{n\times t}_r$ is that it is {\em orthogonally invariant}, meaning that if $X \in \ensuremath{\mathbb{R}}^{n \times t}_r$ then for all orthogonal matrices $U$ and $V$ of appropriate sizes,
$UXV^\top$ is also in $\ensuremath{\mathbb{R}}^{n \times t}_r$. Alternately, membership of $X$ in the set $\ensuremath{\mathbb{R}}^{n\times t}_r$
is determined solely by the vector of singular values $\sigma(X) = (\sigma_1(X),\ldots,\sigma_n(X))$.
Indeed, a matrix $X$ lies in $\ensuremath{\mathbb{R}}^{n\times t}_r$ if and only if its vector of singular values $\sigma(X)$ lies in the set
$${\mathbb{R}}^n_r:=\{x\in{\mathbb{R}}^n: \textup{rank}(x)\leq r\},$$
where $\textup{rank}(x)$ denotes the number of nonzero coordinates of $x$. Observe that the geometry of the piecewise linear set ${\mathbb{R}}^n_r$ is much simpler than that of the highly nonlinear set ${\mathbb{R}}^{n\times t}_r$. In particular, $\ensuremath{\mathbb{R}}^n_r$ is also a variety and $\textup{EDdegree}(\ensuremath{\mathbb{R}}^{n}_r) = {n \choose r}$.
The equality $\textup{EDdegree}(\ensuremath{\mathbb{R}}^{n}_r)=\textup{EDdegree}(\ensuremath{\mathbb{R}}^{n\times t}_r)$ is not accidental;
in this paper, we elucidate this phenomenon.

As alluded to above, our focus in this paper is on a (naturally defined) real analog of critical points for \eqref{prob:ED}. Namely, we say that $x\in{\mathbb{R}}^n$ is a (real) {\em ED critical point} of $y\in{\mathbb{R}}^n$ relative to a set $\mathcal{V}\subset{\mathbb{R}}^n$ (not necessarily a variety) if $x$ is a smooth point of $\mathcal{V}$ and $y-x$ lies in the normal space to $\mathcal{V}$ at $x$; see Definition~\ref{def:real critical points} for details.
Our main result shows that one can always obtain the ED critical points of an orthogonally invariant matrix set by restricting to diagonal matrices, or equivalently, to an {\em absolutely symmetric} set obtained from the singular values of the matrices.
In the last section, we  explore the connection between ED critical points and the critical points in the sense of \cite{DHOST} that happen to be real, a surprisingly subtle topic. In particular, ED critical point calculations often help to understand $\textup{EDdegree}(\mathcal{V})$, when $\mathcal{V}$ is a  variety.

Sets constrained via their singular values are numerous in applications.
Define ${\mathcal E} \subseteq \ensuremath{\mathbb{R}}^{3 \times 3}$ to be the set of rank deficient matrices with two equal singular values:
\begin{equation}\label{eqn:essential}
\mathcal{{\mathcal E}} :=\{X\in{\mathbb{R}}^{3\times 3}: \sigma_1(X)=\sigma_2(X), \,\sigma_3(X)=0\}.
\end{equation}
 A matrix $X \in {\mathcal E}$ is called an {\em essential matrix} in 3D computer vision and represents a pair of calibrated pinhole cameras \cite[Chapter 9]{hartley-zisserman}. The set ${\mathcal E}$ also happens to be a real variety cut out by the following ten cubic equations in the entries of $X$\cite[Proposition 4]{maybank}:
		\begin{align} \label{demazure cubics}
			\det(X) = 0, \,\,\, 2 XX^\top X - {\rm tr}(XX^\top) X = 0.
		\end{align}
Observe that ${\mathcal E}$ is orthogonally invariant as membership of a matrix $X$ in ${\mathcal E}$ depends only on its singular values. We will show that
one can obtain the ED critical points of ${\mathcal E}$ from the simpler set $E_{3,2}$ of vectors in ${\mathbb{R}}^3$ with one coordinate zero and the other two equal in absolute value.

The idea of studying orthogonally invariant matrix sets $ {\mathcal M} \subseteq {\mathbb{R}}^{n\times t}$ via their diagonal restrictions $S=\{x:\mbox{\rm Diag}\, x\in {\mathcal M}\}$ is not new, and goes back at least to von Neumann's theorem on {\em unitarily invariant matrix norms} \cite{von_Neumann}. In recent years, the general theme has become clear:
 various analytic properties of ${\mathcal M}$ and $S$ are in one-to-one correspondence. This philosophy is sometimes called the ``transfer principle''; see for instance, \cite{man}. For example, ${\mathcal M}$ is $C^p$-smooth around a matrix $X$ if and only if $S$ is $C^p$-smooth around $\sigma(X)$ \cite{spec_id,der,diff_2,diff_1}. Other properties, such as {\em convexity} \cite{cov_orig}, {\em positive reach} \cite{spec_prox}, {\em partial smoothness} \cite{spec_id}, and {\em Whitney conditions} \cite{mather} follow the same paradigm. We note in passing that the setting of eigenvalue constrained sets of nonsymmetric matrices is more complicated; see e.g. \cite{math_e-val,nonlip_eig}.  In the current work, we
 derive a transfer theorem for the ED critical points of an orthogonally invariant matrix set ${\mathcal M}$ and illustrate it on several examples. The manuscripts \cite{send,eval,spec_prox} play a central role in our work. The transfer paradigm has many useful features. The first is that in many instances, the calculation over the set $S \subseteq \ensuremath{\mathbb{R}}^n$ is simpler than the original one over ${\mathcal M}$. This can make the formulas for the number of ED critical points of ${\mathcal M}$ much more transparent as compared to the direct calculation in matrix space. The case of $\ensuremath{\mathbb{R}}^{n\times t}_r$ is an example of this. Secondly, our method focuses on (real) ED critical points as opposed to all complex critical points in the setting of algebraic varieties. This is useful in applications and allows us to gauge the difference between the real and complex situations (under appropriate conditions).

This paper is structured as follows. In Section~\ref{sect:ED critical points} we establish basic notation and define the key notion of ED critical points of a set in $\ensuremath{\mathbb{R}}^n$ with respect to a data point $y$. Then in Section~\ref{sect:real critical points of spectral sets} we derive our main theorem
that transfers the study of ED critical points of an orthogonally invariant set of matrices to that of its diagonal restriction.
Section~\ref{sect:applications} illustrates the technique on some concrete examples deriving formulas for the number of ED critical points in each case. Finally in Section~\ref{sect:eddegree} we establish the relationships between our work and the results in \cite{DHOST}, by restricting to algebraic varieties. In particular, we compare our formulas to those for EDdegree in several instances. We note in passing, that many of the results in the paper also hold for
complex matrices with respect to the standard Hermitian inner product.

\section{ED critical points of subsets of $\ensuremath{\mathbb{R}}^n$} \label{sect:ED critical points}
Throughout this paper we consider the $n$-dimensional Euclidean space ${\mathbb{R}}^n$, with a fixed orthonormal basis.
Let $\langle\cdot,\cdot \rangle$ denote the inner product in this setting, and $\|\cdot\|$ denote the induced norm. The {\em distance} and the {\em projection} of a (data) point $y \in \ensuremath{\mathbb{R}}^n$ onto a subset $S \subseteq {\mathbb{R}}^n$, are defined by
 \begin{align*}
 {\rm dist}_S(y)&:= \inf_{x\in S} \|y-x\|, \qquad\textup{ and } \\
 {\rm proj}_S(y) &:= \{x \in S\, : \, {\rm dist}_S(y) = \|y-x\|\}.
 \end{align*}
Computing the distance of $y$ to $S$  amounts to solving the optimization problem:
\begin{align} \label{eq:distance prob}
\inf_{x\in S} \frac{1}{2}\|y-x\|^2 = \inf_{x\in S} \frac{1}{2} \sum_{i=1}^n (y_i-x_i)^2.
\end{align}
A classical first-order necessary condition for a putative
point $x\in S$ to be optimal for \eqref{eq:distance prob} is that the gradient of the object function, namely $x-y$, makes an acute angle with every vector $v$ in the {\em tangent cone}\footnote{If $x$ is an isolated point of $S$, then $\mathcal{T}_{S}(x)$ is declared to consist only of the origin.} $$\mathcal{T}_{S}(x):={\mathbb{R}}_+\Big\{\lim_{z_i\to x} \frac{z_i-x}{\|z_i-x\|}: z_i\in S \Big\}.$$
One can regard such points as generalized critical points of the distance minimization problem \eqref{eq:distance prob}. On the other hand, in order to compare and unify our work with that in \cite{DHOST},  we will impose an extra smoothness condition on a point $x \in S$ in order for it to be considered critical for \eqref{eq:distance prob}. To this end, throughout the manuscript, we fix $p\in \{2, 3, \ldots, \infty, \omega\}$, and say that a point $x \in S$ is {\em $C^p$-smooth} if
there is a neighborhood $\Omega$ of $x$ such that
$S\cap \Omega$ is an embedded $C^p$-smooth manifold
(recall that $C^\omega$ means {\em real analytic}). In this case, the tangent cone $\mathcal{T}_{S}(x)$ is the usual tangent space in the sense of differentiable manifolds, and the criticality condition above amounts to the inclusion $y-x\in {\mathcal N}_{S}(x)$, where ${\mathcal N}_{S}(x)$ denotes the {\em normal space} to $S$ at $x$ --- the orthogonal complement of $\mathcal{T}_{S}(x)$.

From now on we abbreviate ``$C^p$-smooth"  to ``smooth". We will use $S^*$ to denote the set of
all smooth points of $S$. Here is then our main definition, attuned to the one considered in \cite{DHOST}.

\begin{definition}[ED critical points] \label{def:real critical points}
	{\rm
Consider a set $S \subseteq \ensuremath{\mathbb{R}}^n$ and a point $y$ in $\ensuremath{\mathbb{R}}^n$. A point $x \in S$ is an {\em ED critical point} of $y$ on $S$ if the following conditions hold:
\begin{enumerate}[(i)]
\item $x\in S^*$, and
\item $y-x \in {\mathcal N}_S(x)$.
\end{enumerate}
The symbol $\textup{C}_S(y)$ will denote the set of all ED critical points of $y$ on $S$, while the cardinality of $\textup{C}_S(y)$ will be denoted by $C^\#_S(y)$.
}
\end{definition}

By definition, all ED critical points of $y$ on $S$ are real and smooth.
The number of ED critical points $C^\#_S(y)$ varies with $y$. For example, consider the parabola $S$ shown in
Figure~\ref{fig:parabola}, with an additional curve called its {\em ED discriminant} or {\em evolute}.
All points $y$ above the evolute have three  ED critical points while below the evolute they have one  ED critical point.
Since $S$ is an algebraic variety, we can compute its EDdegree which is three, i.e., $S_{\mathbb C}$ has
three distinct regular complex critical points almost everywhere. We will comment more on the ED discriminant in
the Appendix.

\begin{figure}
\includegraphics[scale=0.3]{parabolaWevolute}
\caption{A parabola in the plane with its evolute or ED discriminant.} \label{fig:parabola}
\end{figure}

Letting $\Theta$ be the collection of all Lebesgue null subsets of ${\mathbb{R}}^n$, the following worst-case measure of criticality arises naturally:
$$
C^\#(S) := \inf_{\Gamma\in \Theta } \sup_{y\in \Gamma^c} C^\#_S(y),
$$
where $\Gamma^c$ is the set complement of $\Gamma$ in the ambient space.
Indeed, intuitively if one believes that any single zero measure set $\Gamma$ can be discarded, then $C^\#(S)$ measures the maximal value of $C^\#_S(y)$ that has a non-negligible chance of being encountered. For our purposes, one could think of $C^\#(S)$ as a real analog of
the EDdegree considered in \cite{DHOST}. We discuss this further in Section~\ref{sect:eddegree}.

We note in passing that this criticality measure can be infinite in pathological situations (e.g. union of countably many co-centric circles in ${\mathbb{R}}^2$). On the other hand, for structured sets, such as those that are semi-algebraic, this number is finite.

\begin{proposition}[ED critical points of semi-algebraic sets]  \label{prop:semialgebraic}
For any semi-algebraic set $S$ in ${\mathbb{R}}^n$, the number $C^\#(S)$ is finite.
\end{proposition}
\begin{proof}
Standard quantifier elimination shows that the set $S^*$ is semi-algebraic. Define the manifolds $M_i:=\{x\in S^*: \dim \mathcal{T}_{S}(x)=i\}$ for $i=1,\ldots,n$. Again, quantifier elimination shows that the normal bundle
$$\Omega :=\bigsqcup_{i} \{(x,v): x\in M_i,\, v\in \mathcal{N}_{M_i}(x)\}$$
is a semi-algebraic subset of ${\mathbb{R}}^n\times{\mathbb{R}}^n$ having dimension $n$. Consider now the mapping $\phi\colon\Omega\to{\mathbb{R}}^n$ defined by $\phi(x,v)= x+v$. Notice that $C_S(y)$ coincides with the projection of the preimage $\phi^{-1}(y)$ onto $x$. For dimensional reasons, there is a full-measure set $D\subseteq{\mathbb{R}}^n$ so that for every $y\in D$ the preimage $\phi^{-1}(y)$ has finite cardinality. Moreover, since preimages of a semi-algebraic map have a uniformly bounded number of connected components (see e.g.
\cite[Theorem 3.12]{minimal}), the quantity $C^{\#}_S(y)$ is uniformly bounded on $D$. The result follows.
\end{proof}

Next we consider ED critical points on a union of finitely many sets.
Consider a finite collection of sets $\{S_i\}_{i\in \mathcal{I}}$ in ${\mathbb{R}}^n$ and define the union $U:=\bigcup_{i\in \mathcal{I}} S_i$. In general, the two sets  $C_{U}(y)$ and $\bigcup_{i\in \mathcal{I}} C_{S_i}(y)$ can be vastly different because of the way the sets $S_i$ intersect.  For instance, think of two half spaces whose union is $\ensuremath{\mathbb{R}}^n$. A simple situation in which more can be said is when locally around each critical point $x\in C_U(y)$, the set $U$ coincides with $S_i$ for some $i\in \mathcal{I}$. In that situation, if $x \in C_U(y)$ then $x$ is also in $C_{S_i}(y)$.
An important situation in this paper is the case of all $S_i$ being affine subspaces. We say that a finite collection of sets $\{S_i\}_{i\in \mathcal{I}}$ in ${\mathbb{R}}^n$ is {\em minimally defined} if no $S_i$ is contained in any $S_j$ for distinct indices $i$ and $j$.

\begin{proposition}[ED critical points of affine complexes] \label{prop:linear}
Consider a finite collection of affine subspaces $\{S_i\}_{i\in \mathcal{I}}$ in $\ensuremath{\mathbb{R}}^n$, that is minimally defined, and let  $U:=\bigcup_{i\in \mathcal{I}} S_i$. Then we have
\begin{equation}\label{eqn:crit eq}
U^*=\{x\in{\mathbb{R}}^n: \textrm{ there exists unique }i\in \mathcal{I} \textrm{ with } x\in S_i\}.
\end{equation}
Consequently for any $y\in {\mathbb{R}}^n$, we have
\begin{align} \label{eq:crit subspace}
{\rm C}_{U}(y) =
\bigsqcup_{i\in \mathcal{I}} \left({\rm proj}_{S_i} (y) \cap U^* \right),
\end{align}
and the equality
$$C^\#(U)= |\mathcal{I}|.$$
\end{proposition}

\begin{proof}
The inclusion $\supseteq$ in  \eqref{eqn:crit eq} follows since the sets $S_i$ are affine. To see the reverse inclusion, observe that the tangent cone to $U$ at any point $x$ coincides with the union
$\bigcup_{i\in \mathcal{I}:\, x\in S_i} (S_i-x)$. For $x \in U^*$, the cone $\mathcal{T}_U(x)$ is itself a linear subspace, and hence,
by the minimality of the collection $\{S_i\}_{i\in \mathcal{I}}$,  there exists a unique index $i\in \mathcal{I}$ satisfying $x\in S_i$. This establishes \eqref{eqn:crit eq}. Equation \eqref{eq:crit subspace} then follows immediately.

To see the last claim,
consider the set
$$Z :=\{y\in{\mathbb{R}}^n: \exists i \in\mathcal{I} \textrm{ with }{\rm proj}_{S_i} (y)\cap U^*=\emptyset\}.$$
We will show that $Z$ is a finite union of proper affine subspaces of ${\mathbb{R}}^n$. To see this, consider
a point $y\in Z$ along with an index $i\in\mathcal{I}$ satisfying
${\rm proj}_{S_i} (y)\cap U^*=\emptyset$. From \eqref{eqn:crit eq}, we conclude that there exists $j\in \mathcal{I}$, distinct from $i$, satisfying ${\rm proj}_{S_i} (y)\cap S_{j}\neq \emptyset$. Thus $y$ lies in the set $ S_i^{\perp}+ (S_i\cap S_{j})$. Since the collection $\{S_i\}$ is minimally defined, the intersection $S_i\cap S_{j}$ has dimension strictly smaller than that of $S_i$. Consequently the affine space  $ S_i^{\perp}+(S_i\cap S_{j})$ has dimension strictly smaller than $n$. Taking the union over all pairs of distinct indices $i,j\in\mathcal{I}$, we deduce that $Z$ is a finite union of proper affine subspaces of ${\mathbb{R}}^n$. Therefore, by \eqref{eq:crit subspace}, $C^\#_U(y)=|\mathcal{I}|$ for all $y \not \in Z$ which proves that $C^\#(U)= |\mathcal{I}|$.
\end{proof}

The following two elementary examples illustrate Proposition~\ref{prop:linear}; these are essentially the piecewise linear examples
alluded to in the introduction. In the next section, we will use them
to obtain the ED critical points of the (nonlinear) matrix sets ${\mathbb{R}}^{n\times t}_r$ and ${\mathcal E}$. In what follows we set $[n]  := \{1,\ldots,n\}$.

\begin{example}[Union of $r$-dimensional coordinate subspaces] \label{ex:rank}
	{\rm
		For $x\in\ensuremath{\mathbb{R}}^n$, define the  rank of $x$, denoted by ${\rm rank}(x)$, to be the number of nonzero coordinates of $x$.
		Fix an integer $r \in [n]$ and recall the set
		\begin{equation} \label{eq:def Rnr}
			\ensuremath{\mathbb{R}}^n_r = \{ x \in \ensuremath{\mathbb{R}}^n \,:\,    \textup{rank}(x)\leq r \}.
		\end{equation}
		Define $\mathcal{I}$ to be the collection of distinct cardinality $r$ subsets of $[n]$. Then
		 $|\mathcal{I}|={n \choose r}$, and
		$$\ensuremath{\mathbb{R}}^n_r=\bigcup_{S\in\mathcal{I}} \textrm{span}\, \{e_i\}_{i\in S},$$
		where $e_i$ denotes the $i$'th coordinate vector in $\ensuremath{\mathbb{R}}^n$.
	 This representation of $\ensuremath{\mathbb{R}}^n_r$ is minimal and therefore,  Proposition \ref{prop:linear} implies the equality
			 $C^\#(\ensuremath{\mathbb{R}}^n_r) = |\mathcal{I}| = {n \choose r}.$}
\end{example}

\begin{example}[$k$ nonzero entries equal up to sign] \label{ex:Enk}
	{\rm
		Fix an integer $k\in [n]$, and
		let $E_{n,k}$ be the set of points in $\ensuremath{\mathbb{R}}^n$ with the property that $k$ of their coordinates are equal in absolute value and the other $n-k$ coordinates are zero. Note that $E_{n,k}$ can be written as the union of $2^{k-1} {n \choose k}$ linear subspaces (minimally defined). Proposition \ref{prop:linear} then implies that
		$C^\#(E_{n,k}) = 2^{k-1} {n \choose k}.$
                     As a special case, the set $E_{3,2}$, which is a union of six lines in $\ensuremath{\mathbb{R}}^3$ (see Figure  \ref{fig:e32}),
                     satisfies $C^\#(E_{3,2}) = 6$.	}
\end{example}

\begin{figure}
\includegraphics[scale = 0.5]{e32}
\caption{the set $E_{3,2}$} \label{fig:e32}
\end{figure}

\section{ED critical points of orthogonally invariant matrix sets}  \label{sect:real critical points of spectral sets}

In this section we describe our main result which yields an elegant technique for counting the ED critical points of orthogonally invariant matrix sets, based on the tools established in \cite{send,eval,spec_prox}. Setting the notation, let ${\mathbb{R}}^{n\times t}$ denote the set of real $n\times t$ matrices, where we assume
without loss of generality that $n \leq t$.
The singular value map $\sigma\colon{\mathbb{R}}^{n\times t}\to{\mathbb{R}}^n$ assigns to each matrix $X \in \ensuremath{\mathbb{R}}^{n \times t}$ the vector of its singular values $\sigma(X) := (\sigma_1(X), \ldots, \sigma_n(X))$ arranged in non-increasing order. The corresponding inverse map is defined as
$$\sigma^{-1}(S):=\{X\in {\mathbb{R}}^{n\times t}: \sigma(X)\in S\} \qquad\textrm{ for any subset } S \textrm{ of } {\mathbb{R}}^n.$$

We will be interested in subsets of $\ensuremath{\mathbb{R}}^{n \times t}$ that are invariant under multiplication on the left and right by orthogonal matrices. In what follows, we let $\mathcal{O}^s$ denote the group of real $s\times s$ orthogonal matrices.

\begin{definition}[Orthogonal invariance]{\hfill \\ }
	{\rm
		A set ${\mathcal M} \subseteq \ensuremath{\mathbb{R}}^{n \times t}$ is {\em orthogonally invariant} if
		$${\mathcal M}= U \mathcal{M} V^\top  \qquad\textrm{ for all }  U\in \mathcal{O}^n \textrm{ and } V\in \mathcal{O}^t.$$}
\end{definition}

\vspace{0.15cm}

An example of an orthogonally invariant matrix set is the essential variety
\begin{equation*}
{\mathcal E}=\{X\in {\mathbb{R}}^{3\times 3} \ : \  \sigma_1(X)=\sigma_2(X),\, \sigma_3(X)=0\}
\end{equation*}
considered in the introduction.
At first sight, ${\mathcal E}$ is a complicated set; it is a highly nonlinear real variety cut out by the cubic polynomials in
\eqref{demazure cubics}.
Since orthogonally invariant matrix sets are precisely those matrix sets for which membership is determined solely by the singular values of its elements, the
restriction of such sets to the subspace of diagonal matrices plays an important role. Our strategy will be to exploit this observation to calculate the ED critical points of orthogonally invariant matrix sets.

A mapping $\pi: [n] \rightarrow \{\pm 1, \ldots, \pm n\}$ is a {\em signed permutation}
if  the assignment $i\mapsto|\pi(i)|$  is a permutation on  $[n]$ in the usual sense.
We let $\Pi_n^{\pm}$ denote the set of signed permutations on $[n]$.
Note that any signed permutation $\pi\in \Pi_n^{\pm}$ induces a linear map $\ensuremath{\mathbb{R}}^n \rightarrow \ensuremath{\mathbb{R}}^n$ which we also denote by $\pi$.

\begin{definition}[Absolute symmetry]
{\rm
A set $S \subseteq \ensuremath{\mathbb{R}}^n$ is  said to be {\em absolutely symmetric} if $$S = \pi S\qquad \textrm{ for all  } \pi\in\Pi_n^{\pm}.$$
For any set $S \subseteq \ensuremath{\mathbb{R}}^n$, we call $\Pi_n^{\pm}  S:=\{ \pi x \,:\, \pi \in \Pi_n^\pm \textup{ and } x \in S \} $ the {\em absolute symmetrization} of $S$.}
\end{definition}

For a vector $x \in \ensuremath{\mathbb{R}}^n$, let  $\mbox{\rm Diag}\, x \in \ensuremath{\mathbb{R}}^{n \times t}$ denote the matrix with $x$ in its principal diagonal
and zeros elsewhere.
If ${\mathcal M} \subseteq \ensuremath{\mathbb{R}}^{n \times t}$ is orthogonally invariant, then the set $\{x\in{\mathbb{R}}^n: \mbox{\rm Diag}\, x\in {\mathcal M}\}$ is absolutely symmetric,
and we have the following basic observation (see e.g.  \cite[Proposition 5.1]{send}).

\begin{theorem} [Diagonal Correspondence] \label{thm:sets}
	A set ${\mathcal M} \subseteq \ensuremath{\mathbb{R}}^{n \times t}$ is orthogonally invariant if and only if there exists an
	absolutely symmetric set $S \subseteq \ensuremath{\mathbb{R}}^n$ such that
	${\mathcal M} = \sigma^{-1}(S)$.
\end{theorem}

Thus the assignment $\sigma^{-1}$ from the family of absolutely symmetric sets in $\ensuremath{\mathbb{R}}^n$ to the
family of orthogonally invariant sets in $\ensuremath{\mathbb{R}}^{n\times t}$ is a bijection.
As discussed in the introduction, it so happens that various analytic properties of absolutely symmetric sets $S$ and orthogonally invariant sets $\sigma^{-1}(S)$ are in one-to-one correspondence; see e.g. \cite{man,von_Neumann,der,diff_2,diff_1,cov_orig,spec_prox,spec_id,mather}. This may seem somewhat surprising since $\sigma$ is a highly nonsmooth mapping; it is the absolute symmetry of the underlying set $S$ that makes up for the fact. As an illustration, we show in Theorem~\ref{thm:variety} how the transfer principle for algebraicity can be used to see
that the set of essential matrices ${\mathcal E}$  is a variety, even without explicitly knowing its defining equations  \eqref{demazure cubics}. The proof is entirely analogous to the symmetric case in \cite[Proposition 1.1]{man}; we provide an argument for completeness.

Recall that a set ${\mathcal V} \subseteq \ensuremath{\mathbb{R}}^n$ is called a {\em real variety} if there exist polynomials $f_1, \ldots, f_s \in \ensuremath{\mathbb{R}}[x_1,\ldots,x_n]$ such that ${\mathcal V} = \{ x \in \ensuremath{\mathbb{R}}^n \,:\, f_1(x) = \ldots = f_s(x) = 0 \}$. Note that if $x \in {\mathcal V}$, then $\sum_{i=1}^{s} f_i^2(x) = 0$, and conversely if a point $x \in \ensuremath{\mathbb{R}}^n$ satisfies $\sum_{i=1}^{s} f_i^2(x) = 0$, then $f_i(x) = 0$ for all $i=1,\ldots,s$, and hence, $x \in {\mathcal V}$. Therefore, ${\mathcal V}$ can be described as the set of real zeros of a single sum of squares polynomial with real coefficients, which we call a defining polynomial of ${\mathcal V}$.

\begin{theorem}[Transfer of algebraicity] \label{thm:variety}
	Suppose $S \subseteq \ensuremath{\mathbb{R}}^n$ is an absolutely symmetric set. Then $\sigma^{-1}(S) \subseteq \ensuremath{\mathbb{R}}^{n \times t}$ is a real variety if and only if $S$ is a real variety.
\end{theorem}
\begin{proof}
	If $\sigma^{-1}(S)$ is a real variety, then
	 $S=\{x\in{\mathbb{R}}^n: \mbox{\rm Diag}\, x\in \sigma^{-1}(S)\}$ is the real variety cut out by the equations defining
	 $\sigma^{-1}(S)$ when $x_{ij}$ are set to zero for all $i \neq j$.
	Suppose conversely that $S$ is a real variety with defining polynomial $f \in \ensuremath{\mathbb{R}}[x_1,\ldots,x_n]$.
	Consider the polynomial $\hat{f}(x) :=\sum_{\pi \in \Pi_n^{\pm}} f^2(\pi x)$.

	Note that $\hat{f}(x)$ is also a defining polynomial of $S$, and hence,
	$\sigma^{-1}(S)$ is the zero level set of $\hat{f} \circ \sigma$. To finish the proof, we just need to show that
	$\hat{f}\circ\sigma(X)$ is a polynomial in the entries of $X$.
	To this end, since $\hat{f}$ is invariant under sign changes, it is easy to see that $\hat{f}$ is a symmetric polynomial in the squares $x^2_1,\ldots, x^2_n$, that is we may write $\hat{f}(x)=g(x^2_1,\ldots, x^2_n)$ for some symmetric polynomial $g$. By the fundamental theorem of symmetric polynomials (see e.g. \cite{symm}) we may write $g$ as a polynomial of elementary symmetric polynomials $\epsilon_1,\ldots, \epsilon_n$. On the other hand, the expressions  $\epsilon_i(\sigma^2_1(X),\ldots, \sigma^2_n(X))$ coincide with the coefficients of the characteristic polynomial of $X^\top X$ and are hence, polynomial expressions in the entries of $X$.
\end{proof}

Another useful illustration of transfer principles concerns the distance to orthogonally invariant matrix sets.
Recall that for a set $S \subseteq \ensuremath{\mathbb{R}}^n$, the {\em distance} and {\em projection} of a point $y \in \ensuremath{\mathbb{R}}^n$
to (respectively, onto) $S$ are defined by
${\rm dist}_S(y):= \inf_{x\in S} \|y-x\|,$ and
${\rm proj}_S(y) := \{x \in S \ : \  {\rm dist}_S(y) = \|y-x\|\}$. (The distance and the projection in the matrix space $\ensuremath{\mathbb{R}}^{n\times t}$ are defined analogously with respect to the {\em Frobenius norm} $\|Y\|:=\sqrt{\sum_{i,j} Y^2_{ij}}$.)
Then for an absolutely symmetric set
$S \subseteq  {\mathbb{R}}^n$, the following holds \cite[Proposition 8]{spec_prox}:
\begin{equation}\label{eqn:dist_est}
{\rm dist}_{\sigma^{-1}(S)}(Y) = {\rm dist}_S(\sigma(Y)).
\end{equation}
This in turn implies the following result, which was essentially proved in \cite[Proposition 8]{spec_prox}, though not formally recorded. We provide a proof sketch for completeness.

\begin{proposition}[Projections onto orthogonally invariant matrix sets] \label{prop:proj transfer}
If $S \subseteq \ensuremath{\mathbb{R}}^n$ is an absolutely symmetric set, then for any matrix $Y\in\ensuremath{\mathbb{R}}^{n\times t}$ , the projection ${\rm proj}_{\sigma^{-1}(S)}(Y)$ is precisely the
set
\begin{align} \label{eq:spectral proj}
\left\{ U (\mbox{\rm Diag}\,  x) V^\top \,:\, U\in \mathcal{O}^n, V\in\mathcal{O}^t \textup{ where }
\begin{array}{l}
Y = U (\textup{Diag } \sigma(Y))V^\top,\\
x \in {\rm proj}_S(\sigma(Y))
\end{array}
\right\}.
\end{align}
\end{proposition}

\begin{proof}
Consider first matrices $U\in \mathcal{O}^n, V\in\mathcal{O}^t$ with $Y = U (\textup{Diag } \sigma(Y))V^\top$ and a vector $x\in {\rm proj}_S(\sigma(Y))$. Define $X:=U(\mbox{\rm Diag}\, x)V^\top$ and observe the equalities:
\begin{align*}
\|X-Y\|=\|x-\sigma(Y)\|
={\rm dist}_S(\sigma(Y))={\rm dist}_{\sigma^{-1}(S)}(Y),
\end{align*}
where the last equality follows from \eqref{eqn:dist_est}.
Hence the inclusion $X\in {\rm proj}_{\sigma^{-1}(S)}(Y)$ is valid, as claimed.
Conversely, for any matrix $X\in {\rm proj}_{\sigma^{-1}(S)}(Y)$ observe
$${\rm dist}_{\sigma^{-1}(S)}(Y)= \|X-Y\|\geq  \|\sigma(X)-\sigma(Y)\|\geq {\rm dist}_S(\sigma(Y)),$$
where the first inequality follows from the Von Neumann's trace inequality ~\cite[p. 182]{HJ}; see also \cite[Theorem 4.6]{send}.
Equation  \eqref{eqn:dist_est} implies equality throughout.
In particular we get $\sigma(X)\in {\rm proj}_S(\sigma(Y))$.
Moreover,
applying the equality characterization in the trace inequality \cite[Theorem 4.6]{send}, we conclude that $X$ and $Y$ admit a simultaneous ordered singular value decomposition, that is,
there exist matrices $U\in \mathcal{O}^n, V\in\mathcal{O}^t$ with $Y = U (\textup{Diag } \sigma(Y))V^\top$ and $X = U (\textup{Diag } \sigma(X))V^\top$.
This completes the proof.
\end{proof}

A consequence of Proposition~\ref{prop:proj transfer}  is a convenient representation of the normal space at a point in an orthogonally invariant manifold. First of, the set $\sigma^{-1}(S)$ is smooth around $X$ if and only if $S$ is smooth around $\sigma(X)$ (see e.g. \cite[Theorem 2.4]{spec_id}). Next recall that at any $C^2$-smooth point $x$ of a set $M$, the following equivalence holds:
$$z\in \mathcal{N}_{M}(x) \quad\Longleftrightarrow\quad \{x\}= {\rm proj}_{M}(x+\lambda z) \quad \textrm{ for some } \quad \lambda>0.$$
Proposition~\ref{prop:proj transfer} and a short computation implies
that at any smooth point $X$ of $\sigma^{-1}(S)$ the following formula holds:
$${\mathcal N}_{\sigma^{-1}(S)}(X) = \left\{ U (\mbox{\rm Diag}\,  z) V^\top \,:\, U\in \mathcal{O}^n, V\in\mathcal{O}^t \textup{ with }
\begin{array}{l}
X = U (\mbox{\rm Diag}\, \sigma(X))V^\top,\\
z \in {\mathcal N}_S(\sigma(X))
\end{array}
\right\}.$$
A more general expression without smoothness assumptions can be found in \cite[Theorem 7.1]{send}.
Using the above facts, we now derive a transfer principle for ED critical points which is the main result of this section. The proof uses the following lemma.

\begin{lemma}[Transfer of Lebesgue null sets]\label{lem:meas}
	Consider an absolutely symmetric set $S\subseteq{\mathbb{R}}^n$. Then $S$ is Lebesgue null if and only if $\sigma^{-1}(S)$ is Lebesgue null.
\end{lemma}

We have placed the proof of the lemma above at the end of the section so as to not stray from the narrative.

\begin{theorem}{\rm ({ED critical points of orthogonally invariant matrix sets})}  \label{thm:spectral_calculus} {\hfill \\ }
	Consider an absolutely symmetric set $S\subseteq {\mathbb{R}}^n$ and a matrix
	$Y\in{\mathbb{R}}^{n\times t}$ along with a singular value decomposition $Y=\overline{U}(\mbox{\rm Diag}\, \sigma(Y))\overline{V}^\top$.
	Suppose moreover, that $Y$ has all distinct singular values.
	Then for the orthogonally invariant set ${\mathcal M}:=\sigma^{-1}(S)$, we have
\begin{equation}\label{eqn:crit_corr}
			{\rm C}_{{\mathcal M}}(Y)=\{\overline{U} \big(\mbox{\rm Diag}\,\omega\big)\overline{V}^\top\ \colon  \
	\omega\in {\rm C}_{S}(\sigma(Y))
	\}.
	\end{equation}
	Consequently, we also obtain the equality $C^\#(\mathcal{M})=C^\#(\mathcal{S})$.
\end{theorem}

\begin{proof}
	Consider a critical point $X\in {\rm C}_{{\mathcal M}}(Y)$. Then there exist orthogonal matrices
	$U$ and $V$, and a vector $z\in {\mathcal N}_{S}(\sigma(X))$ satisfying
	$$X=U (\mbox{\rm Diag}\, \sigma(X)) V^\top \quad \textrm{ and }\quad Y=X+U (\mbox{\rm Diag}\, z) V^\top.$$
	Hence we deduce
	$$Y =U (\mbox{\rm Diag}\, (\sigma(X) +z))V^\top.$$
	By uniqueness of singular values and singular vectors, there exists
	a signed permutation matrix $\pi\colon{\mathbb{R}}^n\to{\mathbb{R}}^n$ satisfying $$\sigma(Y)=\pi(\sigma(X)) +\pi(z).$$
	On the other hand, it is easy to verify the inclusion
	$\pi(z)\in{\mathcal N}_{S}(\pi \sigma(X))$.
	Therefore, we obtain $\pi(\sigma(X))\in C_S(\sigma(Y))$.

	Define now the extended signed permutation matrix $\hat{\pi}:=\left[\begin{matrix}
	\pi & 0\\
	0 & I
	\end{matrix}\right] \in {\mathbb{R}}^{t\times t}$.
	Observe then the equality
	$$Y =U (\mbox{\rm Diag}\, \pi^{-1}(\sigma(Y)))V^\top=(U\pi) (\mbox{\rm Diag}\, \sigma(Y))(V\hat{\pi})^\top.$$
	Changing the left and the right singular vector pairs in $U$ and in $V$ by the same sign change in the first place, if needed, we can ensure $U\pi=\overline{U}$ and $V\hat{\pi}=\overline{V}$. Hence,
	$$X=U (\mbox{\rm Diag}\, \sigma(X)) V^\top=\overline{U} (\mbox{\rm Diag}\, \pi(\sigma(X))) \overline{V}^\top,$$
	as claimed.

	Now to prove the reverse inclusion, consider a point $\omega\in {\rm C}_S(\sigma(Y))$.
	Thus there exists a vector $z\in {\mathcal N}_S(\omega)$ satisfying
	$$\sigma(Y)-\omega=z.$$
	Choose a signed permutation $\pi$ so that $\pi(\omega)$ is nonnegative and nonincreasing. Observe then that
	$$\pi(\sigma(Y))-\pi(\omega)=\pi(z)\in {\mathcal N}_S(\pi(\omega)).$$
	Defining $$X:=\overline{U} (\mbox{\rm Diag}\, \omega) \overline{V}^\top=(\overline{U}\pi) (\mbox{\rm Diag}\, \pi(\omega)) (\overline{V}\hat{\pi})^\top,$$ we deduce
	$$X+(\overline{U}\pi) (\mbox{\rm Diag}\, \pi(z)) (\overline{V}\hat{\pi})^\top=(\overline{U}\pi)  (\mbox{\rm Diag}\,\pi(\sigma(Y))) (\overline{V}\hat{\pi})^\top=Y.$$
	The inclusion $X\in {\rm C}_S(Y)$ follows. This establishes equation \eqref{eqn:crit_corr}.

	Define now the set
	$$\Delta:=\{x\in{\mathbb{R}}^n: |x_i|=|x_j| \textrm{ for some indices } i\neq j\}.$$
	Consider a Lebesgue null set $\Gamma\subseteq{\mathbb{R}}^n$. Then clearly the absolutely symmetric set $\widehat{\Gamma}:=\Delta\cup\Pi^{\pm}_n\Gamma$ is Lebesgue null as well. Lemma~\ref{lem:meas} shows that $\sigma^{-1}(\widehat{\Gamma})$ is also Lebesgue null.
Note  for all $\pi\in\Pi^{\pm}_{n}$ and $y\in{\mathbb{R}}^n$, one has $C_S(\pi y)=\pi C_S(y)$ and hence
$C^\#_S(\pi y) = C^\#_S(y)$. Combining these observations, we obtain
	$$C^\#(\mathcal{M})\leq \sup_{Y\in \sigma^{-1}({\widehat{\Gamma}}^c)} C^\#_{\mathcal{M}}(Y)=\sup_{y\in {\widehat{\Gamma}}^c} C^\#_{S}(y)\leq \sup_{y\in \Gamma^c} C^\#_{S}(y),$$
	where the middle equality follows from equation \eqref{eqn:crit_corr}.
	Since $\Gamma$ was an arbitrary null set, we deduce $C^\#(\mathcal{M})\leq C^\#(S)$.

	To see the reverse inequality, consider a Lebesgue null set $\Gamma\subseteq{\mathbb{R}}^{n\times t}$.
           Consider the following subset of $\Gamma$: $$\widehat{\Gamma}:=\{Y\in \Gamma : UYV^\top\in \Gamma \textrm{ for all } U\in\mathcal{O}^n, V\in\mathcal{O}^t\}.$$
	Clearly $\widehat{\Gamma}$ is an orthogonally invariant Lebesgue null set.
	Now trivially we have
	$$\sup_{Y\in\widehat{\Gamma}^c} C^\#_{\mathcal{M}}(Y)\geq \sup_{Y\in\Gamma^c} C^\#_{\mathcal{M}}(Y).$$
	On the other hand, it is easy to verify the equation $C_{\mathcal{M}}(UYV^\top)=U(C_{\mathcal{M}}(Y))V^\top$
and thus, $C^\#_{\mathcal{M}}(UYV^\top) = C^\#_{\mathcal{M}}(Y)$ for all $Y\in{\mathbb{R}}^{n\times t}$ and $U\in\mathcal{O}^n,V\in\mathcal{O}^t$. Hence
	$$\sup_{Y\in\widehat{\Gamma}^c} C^\#_{\mathcal{M}}(Y) =\sup_{Y\in\Gamma^c} C^\#_{\mathcal{M}}(Y).$$
	Notice $\widetilde{\Gamma}:=\widehat{\Gamma}\cup \sigma^{-1}(\Delta)$ is also a Lebesgue null orthogonally invariant set. Hence we deduce
	$$\sup_{Y\in\Gamma^c} C^\#_{\mathcal{M}}(Y)\geq \sup_{Y\in \widetilde{\Gamma}^c} C^\#_{\mathcal{M}}(Y)=\sup_{y\in \sigma(\widetilde{\Gamma})^c} C^\#_{S}(y)\geq C^\#(S),$$
	where the middle equality follows from equation
	\eqref{eqn:crit_corr} and last inequality follows from Lemma~\ref{lem:meas}. Since $\Gamma$ was an arbitrary Lebesgue null set, we deduce the reverse inequality $C^\#(\mathcal{M})\geq C^\#(S)$, and hence equality, as claimed.
\end{proof}

The assumption that singular values of the data point $Y$ are distinct is essential
 in Theorem~\ref{thm:spectral_calculus} as the following example shows.

\begin{example}[Nondistinct singular values]
	{\rm
		Consider the orthogonally invariant set ${\mathcal M} = \{X\in \ensuremath{\mathbb{R}}^{2\times 2} \ : \  \det(X) = 0\}$. We may
		write ${\mathcal M}=\sigma^{-1}(S)$ where $S:=\{ (x_1,x_2)\in \ensuremath{\mathbb{R}}^2  \ : \   x_1 x_2  =0\}$ is absolutely symmetric.
		Then ${\rm C}_S((1,1)) = \{ (1,0), (0,1) \}$. On the other hand, one can verify that
		$uu^\top\in {\rm C}_{{\mathcal M}}(I_2)$ for each $u\in{\mathbb{R}}^2$ of norm one, where $I_2$ is the $2\times 2$ identity matrix.
	}
\end{example}

We now apply Theorem~\ref{thm:spectral_calculus} to our two running examples.

\begin{example}[Matrices of rank at most $r$]
{\rm Fix $r\leq n$ and recall the orthogonally invariant set
$$
\ensuremath{\mathbb{R}}^{n\times t}_r = \{X\in \ensuremath{\mathbb{R}}^{n\times t} \ : \  \textup{rank}(X) \leq r\}.
$$
This set is a determinantal variety cut out by the $(r+1)\times (r+1)$ minors of a symbolic $n\times t$ matrix.
One sees that
$\sigma^{-1}(\ensuremath{\mathbb{R}}^n_r) = \ensuremath{\mathbb{R}}^{n\times t}_r$
where $\ensuremath{\mathbb{R}}^n_r$ is the absolutely symmetric set defined in \eqref{eq:def Rnr}.
Theorem \ref{thm:spectral_calculus} and Example~\ref{ex:rank} together imply that
$$
C^\#(\ensuremath{\mathbb{R}}^{n\times t}_r) = C^\#(\ensuremath{\mathbb{R}}^n_r) = {n \choose r}.
$$
This number is also the ED degree of $\ensuremath{\mathbb{R}}^{n\times t}_r$ (cf. \cite[Example 2.3]{DHOST}); we will revisit this point in Section~\ref{sect:eddegree}. In particular, the ED critical points of a general data matrix $Y$, with singular value decomposition
$Y = U (\mbox{\rm Diag}\, \, \sigma(Y))V^\top$, are the ${n \choose r}$ matrices obtained by setting to zero all possible choices of
$n-r$ singular values in this singular value decomposition. The matrix
$$U \  (\mbox{\rm Diag}\, (\sigma_1(Y), \ldots, \sigma_r(Y), \underbrace{0, \ldots, 0}_{n-r})) \  V^\top $$ is a  nearest element of $\ensuremath{\mathbb{R}}^{n\times t}_r$ to $Y$.
This is precisely the statement of the classical {\em Eckart-Young theorem}.
}
\end{example}

\begin{example}[Essential variety]  \label{ex:essential variety}
{\rm
The essential variety ${\mathcal E}$ defined in \eqref{eqn:essential}
is  orthogonally invariant with
$
{\mathcal E} = \sigma^{-1}(E_{3,2})
$
where $E_{3,2}$ is the absolutely symmetric set defined in Example \ref{ex:Enk}.
Then by Theorem \ref{thm:spectral_calculus} and Example~\ref{ex:Enk},
$$
C^\#({\mathcal E}) = C^\#(E_{3,2}) = 6.
$$
Furthermore, for a matrix $Y\in \ensuremath{\mathbb{R}}^{3\times 3}$ along with a singular value decomposition
$
Y = U (\mbox{\rm Diag}\, \sigma(Y)) V^\top,
$
by (\ref{eq:spectral proj}), the matrix
$$
U \ \left(\mbox{\rm Diag}\, \left( \frac{\sigma_1(Y)+\sigma_2(Y)}{2}, \frac{\sigma_1(Y)+\sigma_2(Y)}{2}, 0\right) \right) \ V^\top
$$
is a nearest element of ${\mathcal E}$ to $Y$.
This is precisely Hartley's result \cite[Theorem 5]{hartley} which is well known in the
computer vision community. The six ED critical points of a general $Y = U (\mbox{\rm Diag}\, \, \sigma(Y))V^\top$ are the matrices
$U(\mbox{\rm Diag}\, \, x)V^\top$ where $x$ varies over the following vectors:\\
\begin{small}
$$
\begin{array}{cc}
 \left( \frac{\sigma_1(Y)+\sigma_2(Y)}{2}, \frac{\sigma_1(Y)+\sigma_2(Y)}{2}, 0\right), &
 \left( \frac{\sigma_1(Y)-\sigma_2(Y)}{2}, \frac{-\sigma_1(Y)+\sigma_2(Y)}{2}, 0\right), \\
  \left( \frac{\sigma_1(Y)+\sigma_3(Y)}{2}, 0, \frac{\sigma_1(Y)+\sigma_3(Y)}{2}\right), &
 \left( \frac{\sigma_1(Y)-\sigma_3(Y)}{2}, 0, \frac{-\sigma_1(Y)+\sigma_3(Y)}{2}\right),\\
 \left( 0,\frac{ \sigma_2(Y)+\sigma_3(Y)}{2}, \frac{\sigma_2(Y)+\sigma_3(Y)}{2}\right), &
 \left(0, \frac{\sigma_2(Y)-\sigma_3(Y)}{2}, \frac{-\sigma_2(Y)+\sigma_3(Y)}{2}\right). \\
\end{array}
$$
\end{small}
}
\end{example}

We end the section with the proof of the lemma evoked in the proof of Theorem~\ref{thm:spectral_calculus}.

\begin{proof}[Proof of Lemma~\ref{lem:meas}]
	Before we get to the main part of the proof, we need some notation. To this end,
	define $\mathcal{O}^{t}_n$ to be the set of $t\times n$ matrices with orthonormal columns.
          Note that $\mathcal{O}^{t}_n$ is a smooth manifold of dimension $nt-\frac{n(n+1)}{2}$, and is called a {\em Stiefel manifold};
          see for example \cite[Proposition A.4]{burgisser}.
          In particular $\mathcal{O}^n$ is a manifold of dimension $\frac{n(n-1)}{2}$.
           We deduce that the product manifold
	$\mathcal{O}^n\times\mathcal{O}^{t}_n\times {\mathbb{R}}^n$ has dimension exactly $nt$, same as ${\mathbb{R}}^{n\times t}$.
	For any vector $x\in{\mathbb{R}}^n$ let $\textrm{sDiag } x$ denote the $n\times n$ diagonal matrix with $x$ on the diagonal.
	Define now the mapping $\Gamma\colon \mathcal{O}^n\times\mathcal{O}^{t}_n\times {\mathbb{R}}^n \to {\mathbb{R}}^{n\times t}$ by setting $\Gamma(U,V,x)=U(\textrm{sDiag } x) V^\top$.
	Notice that for the absolutely symmetric set $S$, we have equality $\Gamma(\mathcal{O}^n\times\mathcal{O}^{t}_n\times S)=\sigma^{-1}(S)$.

	Suppose now that $S$ is a Lebesgue null set. Then clearly the set $\mathcal{O}^n\times \mathcal{O}^{t}_n\times S$ is Lebesgue null in the manifold $\mathcal{O}^n\times\mathcal{O}^{t}_n\times {\mathbb{R}}^n$. Since $\Gamma$ is smooth, the image $\Gamma(\mathcal{O}^n\times \mathcal{O}^{t}_n\times S)$, which coincides with $\sigma^{-1}(S)$, is Lebesgue null ${\mathbb{R}}^{n\times t}$, as claimed.
	Conversely, suppose that $\sigma^{-1}(S)$ is Lebesgue null.
	It is well-known that $\Gamma$ is a local diffeomorphism on an open full-measure subset
	of the domain manifold $\mathcal{O}^n\times \mathcal{O}^{t}_n\times {\mathbb{R}}^n$.
	To see this quickly, define the set
	$$\Delta=\{x\in{\mathbb{R}}^{n}: |x_i|= |x_j| \textrm{ for some indices }i\neq j\}.$$
	Observe that since $\Gamma$ is a semi-algebraic map, we may stratify $\mathcal{O}^n\times \mathcal{O}^{t}_n\times \Delta^c$ into finitely many smooth manifolds ${\mathcal{M}_i}$ so that the restriction of $\Gamma$ to each $\mathcal{M}_i$ has constant rank. Suppose now for the sake of contradiction that the derivative of $\Gamma$ has deficient rank on some maximal dimensional manifold $\mathcal{M}_i$. Then by the constant rank theorem \cite[Theorem 5.13]{Lee2},
	there exists a nontrivial smooth path $(U(t),V(t),x(t))$ in $\mathcal{M}_i$
	so that	$\Gamma$ is constant on the path, meaning that the matrices $U(t)(\textrm{sDiag } x(t)) V(t)^\top$ are equal for all $t$. On the other hand, taking into account that the coordinates of $x(t)$ are distinct and appealing to uniqueness of singular values and singular vectors, we obtain a contradiction.
Hence $\Gamma$ is a local diffeomorphism on an open full measure subset of $\mathcal{O}^n\times \mathcal{O}^{t}_n\times {\mathbb{R}}^n$.
It follows immediately that $S$ is Lebesgue null, since otherwise the image $\Gamma(\mathcal{O}^n\times \mathcal{O}^{t}_n\times S)$ would fail to be Lebesgue null. This completes the proof.

\end{proof}

\section{Applications} \label{sect:applications}

We now apply the techniques developed in the last section to calculate and count the ED critical points of several orthogonally invariant matrix sets.

\begin{example}[Matrices orthogonally equivalent to a given matrix]   \label{ex:OE}
{\rm
Fix a matrix $A\in \ensuremath{\mathbb{R}}^{n\times t}$. We say $X\in \ensuremath{\mathbb{R}}^{n\times t}$ is
{\em orthogonally equivalent} to $A$ if $X = UAV^\top$ for some $U\in \mathcal{O}^n$ and $V\in \mathcal{O}^t$.
Let $\mathcal{M}_A\subseteq \ensuremath{\mathbb{R}}^{n\times t}$ be the set of matrices that are orthogonally equivalent to $A$.
Then ${\mathcal M}_A$ is orthogonally invariant and ${\mathcal M}_A= \sigma^{-1} (\sigma(A))$. By
Theorem \ref{thm:spectral_calculus},
$$C^\#(\mathcal{M}_A) = |\Pi_n^\pm \{\sigma(A)\}| = 2^n \frac{n!}{n_1! \ldots n_k!}$$
where $\sigma(A) = (\underbrace{\sigma_1,\ldots,\sigma_1}_{n_1}, \ldots,\underbrace{\sigma_k,\ldots,\sigma_k}_{n_k} )$.
}
\end{example}

\begin{example}[Real orthogonal group] \label{ex:On}
{\rm
The real orthogonal group $\mathcal{O}^n$ is a special case of the previous example
because $\mathcal{O}^n$ is the set of $n\times n$ real matrices that are orthogonally equivalent to
the $n\times n$ identity matrix, namely,
$$\mathcal{O}^n = \sigma^{-1} ((\underbrace{1,\ldots,1}_{n})).$$
Hence $C^\#(\mathcal{O}^n) = 2^n$ which is also the ED degree of $\mathcal{O}^n$
(cf. \cite[Theorem 3.2]{draisma}).

By Proposition~\ref{prop:proj transfer}, for any $Y\in \ensuremath{\mathbb{R}}^{n\times n}$ with singular value decomposition
$Y  = U (\mbox{\rm Diag}\, \sigma(Y)) V^\top$ where  $U,V\in \mathcal{O}^n$,
one has $UV^\top \in {\rm proj}_{\mathcal{O}^n} (Y)$. Moreover, if the singular values of $Y$ are distinct, then by
Theorem~\ref{thm:spectral_calculus}, the set of
critical points of $Y$ on $\mathcal{O}^n$ is $\{ U ( {\rm Diag } \,x)V^\top \,:\, x_i = \pm 1 \,\,\forall \,\,i \}$.
These results were also obtained by Draisma and Baaijens \cite{draisma} using algebraic methods.
}
\end{example}

\begin{example}[Unit sphere of Schatten $d$-norm]  \label{ex:unit ball}
{\rm
The Schatten $d$-norm of a matrix $X\in \ensuremath{\mathbb{R}}^{n\times t}$ is defined as
$$
\|X\|_d := \left[  \sum_{i=1}^n (\sigma_i (X))^d  \right]^{1/d}.
$$
Note the following fact: $\|U X V^\top\|_d = \|X\|_d$ for any $X\in \ensuremath{\mathbb{R}}^{n\times t}$, $U\in \mathcal{O}^n$,
$V \in \mathcal{O}^t$.
Hence the unit sphere of the Schatten $d$-norm
$$
\mathcal{F}_{n,t,d} := \{X\in \ensuremath{\mathbb{R}}^{n\times t} \ : \   \|X\|_d = 1\} $$
is orthogonally invariant. Assume $d\geq 2$ is an even integer. One has
$
\mathcal{F}_{n,t,d} = \sigma^{-1}(F_{n,d})
$
where $F_{n,d}$ is the {\em affine Fermat hypersurface}
\begin{align*}
F_{n,d} := \left\{x\in \ensuremath{\mathbb{R}}^n \ : \  \sum_{i=1}^n x_i^d = 1\right\}
\end{align*}
which is absolutely symmetric. Therefore, $C^\#(\mathcal{F}_{n,t,d}) = C^\#(F_{n,d})$ for all $t$.

\begin{figure}
\centering
\includegraphics[scale = 0.35]{fermat_simple}
\caption{The curves $F_d$ and $\gamma_d$ in Example \ref{ex:unit ball} for $d=4$} \label{fig:fermat}
\end{figure}

Consider the special case $n=2$.  For simplicity, we set $F_d := F_{2,d}$. Then $F_2$ is the unit circle
and $C^\#(F_{2}) = 2$. Suppose $d\geq 4$.
For any $y\in \ensuremath{\mathbb{R}}^2$, a point $x$ is an ED critical point of $y$ on $F_d$ if and only if
$x$ lies on both $F_d$ and the curve
\begin{align} \label{eq:fermat system}
 \gamma_d:= \{x\in \ensuremath{\mathbb{R}}^2 \ : \  x_1^{d-1}(x_2-y_2) = x_2^{d-1}(x_1-y_1)\}.
\end{align}
The graph of $\gamma_4$ is shown in Figure \ref{fig:fermat}.
By symmetry, we may assume that $y_1>y_2>0$.
When $y_1$ is small, the ``optimal'' points, $m_1,m_2,m_3,m_4$,
on the pieces of $\gamma_d$ in the coordinate directions lie inside the curve $F_d$, namely,
$$m_i \in \{x\in \ensuremath{\mathbb{R}}^2 \ : \  x_1^d + x_2^d <1\}$$ for $i =1,\ldots,4$.
Hence,
$C^\#_{F_{d}}(y) \leq 8$ for any point $y$ except on a null set, and
there is an open set $V$ in $\ensuremath{\mathbb{R}}^2$ such that $C^\#_{F_{d}}(y) = 8$ for any $y\in V$.
To sum up, by Theorem \ref{thm:spectral_calculus}, we have
\begin{align*}
C^\# (\mathcal{F}_{2,t,d}) = C^\#(F_d) =
\begin{cases}
2 & \mbox{ if } d =2\\
8 & \mbox{ if } d\geq 4.
\end{cases}
\end{align*}
}
\end{example}

\begin{example}[$SL_n^\pm$]  \label{ex:special linear}
{\rm
This example was considered in \cite[Section 4]{draisma}. Define the orthogonally invariant set
$$
SL_n^\pm := \{X\in \ensuremath{\mathbb{R}}^{n\times n} \ : \   \det (X) = \pm 1\} = \sigma^{-1}(H_n)
$$
where  $H_n$ is the absolutely symmetric set defined as
\begin{align*}
H_n := \{x\in \ensuremath{\mathbb{R}}^n \ : \  x_1 \cdots x_n = \pm 1\}.
\end{align*}
Consider the special case $n=2$.
We evaluate $C^\# (SL^\pm_2)$ by computing $C^\#(H_2)$.
The set $H_2$ is a disjoint union of two hyperbolas:
$$
H_2^\pm := \{ x\in \ensuremath{\mathbb{R}}^2 \ : \  x_1x_2 = \pm 1\}.
$$
Since these hyperbolas are disjoint, for any $y$, one has
$$
C^\#_{H_2}(y) = C^\#_{H_2^+}(y) + C^\#_{H_2^-}(y).
$$
Notice that each of the sets ${\rm C}_{H_2^\pm}(y)$ is defined by the roots of
a univariate quartic. Indeed,
$$ C_{H_2^+}(y) =  \left\{ \left(x,\frac{1}{x} \right) \in \ensuremath{\mathbb{R}}^2 \ : \ q_y^+(x) := x^4- x^3 y_1 + xy_2-1 = 0 \right\}$$
and
$$ C_{H_2^-}(y) = \left\{ \left(x, \frac{-1}{x} \right) \in \ensuremath{\mathbb{R}}^2 \ : \ q_y^-(x) := x^4- x^3 y_1 - xy_2-1 = 0 \right\}.$$
By computing the trace forms of these quartics and applying Sylvester's criterion (see e.g. \cite[Corollary 2.9]{sturmfels2002})
we acquire these results:
$$
C^\#_{H_2}(y) =
\begin{cases}
6 & \mbox{ if } D^+(y)>0 \text{ or } D^-(y)>0\\
4 & \mbox{ if } D^+(y)<0 \textup{ and } D^-(y)<0
\end{cases}
$$
where the bivariate polynomials
$$D^+(y) := -256 + 192y_1 y_2  + 6 y_1^2 y_2^2
+ 4y_1^3 y_2^3   -27 y_1^4   - 27 y_2^4$$
and
$$D^-(y) := -256 - 192y_1 y_2  + 6 y_1^2 y_2^2
- 4y_1^3 y_2^3   -27 y_1^4   - 27 y_2^4$$
are the discriminants of $q^+_y$ and $q^-_y$ respectively; see Figure \ref{fig:chamber}.
Then by Theorem \ref{thm:spectral_calculus}, we know
$$C^\# (SL^\pm_2) = 6.$$
}
\end{example}

\begin{figure}
\includegraphics[scale = 0.5]{dpm}
\caption{Values of $C^\#_{H_2}(y)$ for general $y$, in Example \ref{ex:special linear}} \label{fig:chamber}
\end{figure}

\section{Connections to the Euclidean distance degree} \label{sect:eddegree}
As mentioned in the introduction, this work was inspired by the general framework in \cite{DHOST} for counting the critical points of the squared Euclidean distance function on an algebraic variety. In this section we comment on the relationships between our results and those in \cite{DHOST}, beginning
with some basic facts about real varieties \cite{whitney,BCR}.

Consider a real variety  ${\mathcal V} \subseteq \ensuremath{\mathbb{R}}^n$ whose vanishing ideal
is generated by the polynomials $f_1,\ldots,f_s \in \ensuremath{\mathbb{R}}[x_1,\ldots,x_n]$.
Then the {\em Zariski closure} of ${\mathcal V}$ is ${\mathcal V}_{\mathbb C} := \{ x \in {\mathbb C}^n \,:\, f_1(x) = \ldots = f_s(x)=0 \}$, the smallest complex variety containing ${\mathcal V}$. The real part of ${\mathcal V}_{\mathbb C}$,
i.e., ${\mathcal V}_{\mathbb C} \cap \ensuremath{\mathbb{R}}^n$, is precisely ${\mathcal V}$ and hence, the vanishing ideal of
${\mathcal V}_{\mathbb C}$ in ${\mathbb C}[x_1,\ldots,x_n]$ is also generated by $f_1, \ldots, f_s$. Also, both ${\mathcal V}$ and ${\mathcal V}_{\mathbb C}$ have the same dimension over ${\mathbb{R}}$ and ${\mathbb{C}}$ respectively.
Recall that if ${\mathcal V}_{\mathbb C} = \bigcup_{j=1}^t \mathcal{W}^j$ is a minimal irreducible decomposition of ${\mathcal V}_{\mathbb C}$,
then for any $j$, the real part of $\mathcal{W}^j$, denoted by $\mathcal{U}^j$, is an irreducible
real variety whose Zariski closure is $\mathcal{W}^j$. Moreover, ${\mathcal V} = \bigcup_{j=1}^t \mathcal{U}^j$
is a minimal irreducible decomposition of ${\mathcal V}$.  For simplicity, in the rest of this section we will assume that ${\mathcal V}_{\mathbb C}$ is irreducible.

Denote by $J(f)$ the $s \times n$ {\em Jacobian} matrix whose $(i,j)$-entry is $\frac{\partial f_i}{\partial x_j}$. Then a point $x \in {\mathcal V}_{\mathbb C}$ is said to be {\em regular}
if the rank of the Jacobian matrix evaluated at $x$, $\textup{rank}(J(f)(x))$, equals the codimension of ${\mathcal V}_{\mathbb C}$. Let ${\mathcal V}_{\mathbb C}^{\textup{reg}}$ denote the regular points of ${\mathcal V}_{\mathbb C}$. It is known that ${\mathcal V}_{\mathbb C}^{\textup{reg}}$ is a Zariski open subset of ${\mathcal V}_{\mathbb C}$, and its complement is a proper subvariety in ${\mathcal V}_{\mathbb C}$, denoted as ${\rm Sing}({\mathcal V}_{\mathbb C})$, and called the {\em singular locus} of ${\mathcal V}_{\mathbb C}$.
The set ${\mathcal V}^{\rm reg}$ of regular points in ${\mathcal V}$ is precisely the set of real points in ${\mathcal V}_{\mathbb C}^{\textup{reg}}$. However,
 while ${\mathcal V}_{\mathbb C}^{\textup{reg}}$ is dense in ${\mathcal V}_{\mathbb C}$ in the Euclidean topology, ${\mathcal V}^{\textup{reg}}$ may not be dense in ${\mathcal V}$ or even
 ${\mathcal V}^*$, the set of smooth (i.e., $C^p$-smooth) points in ${\mathcal V}$. The following example illustrates this behavior.

 \begin{example} [Cartan umbrella] \label{ex:cartan}
{\rm The {\em Cartan umbrella} is the real variety
$$ {\mathcal V} = \{ (x_1,x_2,x_3) \in \ensuremath{\mathbb{R}}^3 \,:\, x_3(x_1^2+x_2^2)-x_1^3=0 \}.$$
Here ${\mathcal V}_{\mathbb C}$ is irreducible, as is ${\mathcal V}$. On the other hand, ${\mathcal V}$ is also the union of a surface (which is not a real variety) and the $x_3$-axis (Figure~\ref{fig:cartanumbrella}).
\begin{figure}
    \includegraphics[scale=0.5]{cartanumbrella}
    \caption{The Cartan umbrella}
    \label{fig:cartanumbrella}
\end{figure}
No point on the $x_3$-axis is regular while all points on the $x_3$-axis, except the origin, are smooth.}
\end{example}

This example shows an important distinction between smooth and regular points on a real variety. Our goal in this section is
to elaborate on this difference in the context of critical points as defined in \cite{DHOST}.
 Recall that the normal space at a regular point $x \in {\mathcal V}_{\mathbb C}$, denoted as  $\mathcal{N}^{\rm alg}_{{\mathcal V}_{\mathbb C}}(x)$, is the row space
of $J(f)(x)$.
Given a data point $y \in {\mathbb C}^n$, and a real variety ${\mathcal V}$, the authors of \cite{DHOST} study
the set of critical points
 \begin{align} \label{eq:alg crit point}
C^{\rm reg}_{{\mathcal V}_{\mathbb C}}(y) := \left\{ x \in {\mathcal V}_{\mathbb C}^{\rm reg} \,:\, y-x \in \mathcal{N}^{\rm alg}_{{\mathcal V}_{\mathbb C}}(x) \right\}.
\end{align}
Note that in the above algebraic setting, even though our input is a real variety ${\mathcal V} \subseteq \ensuremath{\mathbb{R}}^n$,
there is a natural passage into the complex numbers; the real variety ${\mathcal V}$ is replaced by the
complex variety ${\mathcal V}_{\mathbb C}$, the data point can be any $y \in {\mathbb C}^n$, and the critical points of
$y$ are regular points in ${\mathcal V}_{\mathbb C}$.
This complexification yields the first key result in \cite{DHOST}, namely that for a fixed ${\mathcal V}$, the number of
critical points of a general data point $y \in {\mathbb C}^n$ is a constant. In other words, the cardinality of $C^{\rm reg}_{{\mathcal V}_{\mathbb C}}(y)$,
which we will denote as $C^{\rm \# \,reg}_{{\mathcal V}_{\mathbb C}}(y)$, is a constant.
This constant is called the {\em Euclidean distance degree (EDdegree)} of ${\mathcal V}$ in \cite{DHOST}. The phrase ``general data point" refers
to the fact that $C^{\rm \# \,reg}_{{\mathcal V}_{\mathbb C}}(y)$ is different from ${\rm EDdegree}({\mathcal V})$ only if $y$ lies on
certain proper subvarieties in ${\mathbb C}^n$ which
we will describe in the Appendix. We remark that if ${\mathcal V}_{\mathbb C}$ is reducible, then ${\rm EDdegree}({\mathcal V})$ is the sum of the ED degrees of its
irreducible components.

Our results in this paper provide a method to compute the (smooth) ED critical points of a real orthogonally invariant set of matrices. We did not require this
set to be a real variety, and the notion of smoothness (of critical points) was geometric
and not algebraic. Therefore, in order to connect our work to that in \cite{DHOST}, we need to restrict to real varieties
${\mathcal V} \subseteq \ensuremath{\mathbb{R}}^n$,  data points $y \in \ensuremath{\mathbb{R}}^n$,
and understand the relationships between the following two sets of critical points of $y$:
\begin{itemize}
\item $C_{{\mathcal V}}(y)$, the set of ED critical points of $y$ on ${\mathcal V}$ as in Section~\ref{sect:ED critical points},
and
\item $C_{{\mathcal V}}^{\rm reg}(y)$, the set of regular critical points of $y$ on ${\mathcal V}$.
\end{itemize}
As usual we denote the cardinalities of these sets by $C^\#_{\mathcal V}(y)$ and $C_{\mathcal V}^{\rm \# \,reg}(y)$ respectively.
Also, note that $C_{{\mathcal V}}^{\rm reg}(y) = C^{\rm reg}_{{\mathcal V}_{\mathbb C}}(y) \cap \ensuremath{\mathbb{R}}^n = C^{\rm reg}_{{\mathcal V}_{\mathbb C}}(y) \cap {\mathcal V}$.
Since a regular point of ${\mathcal V}$ is $C^p$-smooth for $p\in \{\infty,\omega\}$,  for any $y$ we have,
\begin{align} \label{eq:reg is contained in smooth}
C_{{\mathcal V}}^{\rm reg}(y) \subseteq C_{{\mathcal V}}(y)\,\,\,\, \textup{ and so,} \,\,\,\, C_{\mathcal V}^{\rm \# \,reg}(y) \leq C^\#_{\mathcal V}(y).
\end{align}
However, for a given $y$, this inequality can be strict since there can be smooth points on ${\mathcal V}$ that are not regular as we saw in the Cartan umbrella.
In fact, in the Cartan umbrella there is a dense set of $y$ for which the inequality in \eqref{eq:reg is contained in smooth} is strict since for any $y$ with $y_3 \neq 0$, there will be an ED critical point on the $x_3$-axis.

Moving on to invariants of the entire set ${\mathcal V}$, on the one hand, we have the quantity
$C^\#({\mathcal V}) = \inf_{\Gamma\in \Theta } \sup_{y\in \Gamma^c} C^\#_{\mathcal V}(y)$ from Section~\ref{sect:ED critical points},
and on the other hand, we have the constant ${\rm EDdegree}({\mathcal V})$ which is an upper bound to
$C_{{\mathcal V}_{\mathbb C}}^{\rm \# \, reg}(y)$ whenever $C_{{\mathcal V}_{\mathbb C}}^{\rm reg}(y)$ is finite.
Since the inequality in \eqref{eq:reg is contained in smooth} can be strict, it is not clear that the number $C^\#({\mathcal V})$ is
always a lower bound to $\textup{EDdegree}({\mathcal V})$.  However, it is true under certain conditions.

\begin{theorem} \label{thm:cV at most EDdegree}
If ${\mathcal V}$ is a real variety such that ${\mathcal V}^{\rm reg}$ is Euclidean dense in ${\mathcal V}^*$, then we have the inequality
$C^\#({\mathcal V})\leq {\rm EDdegree}({\mathcal V})$.
\end{theorem}

This is an immediate consequence of the following lemma. Note that the Cartan umbrella does not satisfy the
assumptions of Theorem~\ref{thm:cV at most EDdegree}.

\begin{lemma}  \label{lem:suff cond}
Consider a real variety ${\mathcal V} \subseteq \ensuremath{\mathbb{R}}^n$ (possibly reducible) such that ${\mathcal V}^{\rm reg}$ is Euclidean dense in ${\mathcal V}^*$.
Then there is an open set of points $y\in \ensuremath{\mathbb{R}}^n$ such that
\begin{align} \label{lem:condition3}
& \text{for any irreducible component $\mathcal{W}$ of ${\mathcal V}$,  }
\text{$ C_\mathcal{W_{\mathbb C}}^{\rm  \# \,reg} (y) = {\rm EDdegree}(\mathcal{W})$,}\\
& C^\#_{\mathcal V}(y) = C^\#({\mathcal V}), \ \text{ and }     \label{lem:condition1} \\
& C_{{\mathcal V}}(y)\subseteq C_{{\mathcal V}}^{\rm reg}(y).\label{lem:condition2}
\end{align}
\end{lemma}

\begin{proof}
Fix an irreducible component $\mathcal{W}$ of ${\mathcal V}$. Then from \cite[Theorem 4.1]{DHOST} the set of all $y \in {\mathbb C}^n$ for which
$C^{\rm \# \, reg}_{\mathcal{W}_{\mathbb C}}(y)\neq {\rm EDdegree}(\mathcal{W})$  lies in a proper subvariety
$D$ in  ${\mathbb C}^n$.
Consider the real variety $D_\ensuremath{\mathbb{R}}:= D\cap \ensuremath{\mathbb{R}}^n$ which contains all the real points $y$ with
$C^{\rm \# \, reg}_{\mathcal{W}_{\mathbb C}}(y)\neq {\rm EDdegree}(\mathcal{W})$.
The Zariski closure of $D_\ensuremath{\mathbb{R}}$ is a subvariety of $D$ with complex dimension equal to  the real dimension of $D_\ensuremath{\mathbb{R}}$.
Therefore, the real dimension of $D_\ensuremath{\mathbb{R}}$ is smaller than $n$, and hence $D_\ensuremath{\mathbb{R}}$ is a proper subvariety in
$\ensuremath{\mathbb{R}}^n$. In particular, the complement of $D_\ensuremath{\mathbb{R}}$ in $\ensuremath{\mathbb{R}}^n$ is an open full measure set. Taking the intersection of all such
open full measure sets as we vary over the irreducible components of ${\mathcal V}$, we still have an open full measure set, and any point
$y$ in this set satisfies \eqref{lem:condition3}.

Since ${\mathcal{V}}$ is a variety, one has $C^\#({\mathcal{V}})<\infty$ by Proposition \ref{prop:semialgebraic}.
Then by quantifier elimination, the set
$\{y\in \ensuremath{\mathbb{R}}^n \ : \   C^\#_{{\mathcal{V}}}(y) = C^\#({\mathcal V})\}$ is semialgebraic. Moreover, this set cannot be Lebesgue null by the definition of $C^\#({\mathcal V})$. Therefore, it must contain an open subset.
Hence \eqref{lem:condition1}
holds for all $y$ in some open set.

We now show that \eqref{lem:condition2} holds for any point $y$ in an open full measure set, which will prove the lemma.
For any $i= 0, 1,\ldots, n$ consider the submanifold
$$
M_i := \{x\in {\mathcal{V}}^* \ : \  {\rm dim}(\mathcal{T}_{{\mathcal V}}(x))=i \}.
$$
These submanifolds $M_i$ clearly partition ${\mathcal{V}}^*$. We first claim $M_i\cap {\rm cl}(M_j)=\emptyset$ for all distinct pairs $i,j \in [n]$.
Note the symbol ``cl" here denotes the Euclidean closure.
Indeed, if there existed some point $x\in M_i \cap {\rm cl}(M_j)$, then we would deduce for all points $y\in M_j$ sufficiently near $x$, the equality $j=\dim \mathcal{T}_{\mathcal{V}}(y)= \dim \mathcal{T}_{\mathcal{V}}(x) =  i$, a contradiction. The middle equality follows from the fact that $\dim \mathcal{T}_{\mathcal V}( \cdot )$ is constant in a neighborhood of $x$ in ${\mathcal V}$.

Now since ${\mathcal{V}}^{\rm reg}$ is dense in ${\mathcal{V}}^*$, the inclusion ${\mathcal V}^* \subseteq {\rm cl}({\mathcal V}^{\rm reg}) =  \bigcup_i {\rm cl}({\mathcal V}^{\rm reg} \cap M_i)$ holds.
We claim that for any $i$, the set ${\mathcal{V}}^{\rm reg} \cap M_i$ is Euclidean dense in $M_i$,
namely, $M_i \subseteq {\rm cl}({\mathcal{V}}^{\rm reg} \cap M_i)$.
To see this, it suffices to establish $M_i \cap {\rm cl}({\mathcal{V}}^{\rm reg}\cap M_j)=\emptyset$ for any $j\neq i$. This follows immediately by observing
$M_i \cap {\rm cl}({\mathcal{V}}^{\rm reg}\cap M_j)\subseteq M_i \cap {\rm cl}(M_j) =\emptyset$.
It follows that for any $i$, the strict inequality
$$
\dim (M_i \setminus ({\mathcal{V}}^{\rm reg}\cap M_i)) < \dim M_i
$$
holds (see \cite[Proposition 3.16]{costeIntrotoSAG}).
By dimension arguments, the dimension of the set
$$
\bigcup_{x\in M_i \setminus (M_i \cap {\mathcal{V}}^{\rm reg})} (x+{\mathcal N}_{\mathcal{V}}(x))
$$
is strictly less than $n$. In particular, its interior is empty.
Taking the union over $i$,
$$
\bigcup_{x\in {\mathcal{V}}^* \setminus {\mathcal{V}}^{\rm reg}}(x+{\mathcal N}_{\mathcal{V}}(x))
$$
also has empty interior.
Any $y$ which is not in this set satisfies \eqref{lem:condition2}.
\end{proof}

We conclude the paper by comparing $C^\#({\mathcal V})$ and ${\rm EDdegree}({\mathcal V})$ in some of the examples
we saw in Section~\ref{sect:real critical points of spectral sets} and Section~\ref{sect:applications}.

\begin{example} [$SL_n^{\pm}$] \label{ex:lb sln}
{\rm
Every point in the  Zariski closure of $SL_n^\pm$ (c.f. Example \ref{ex:special linear})
is regular. Therefore by Theorem \ref{thm:cV at most EDdegree}, we obtain
$$
C^\#(SL_n^\pm) \leq {\rm EDdegree}(SL_n^\pm) = n2^n.
$$
The formula for EDdegree was derived in \cite{draisma}.
We saw in Example \ref{ex:special linear} that  strict inequality holds when $n=2$.
}
\end{example}

\begin{example} [Unit sphere of Schatten $d$-norm] \label{ex:lb fermat}
{\rm
Consider the set $\mathcal{F}_{2,2,d} \subseteq \ensuremath{\mathbb{R}}^{2\times 2}$ given in Example \ref{ex:unit ball}.
For any $d$  the Zariski closure of   $\mathcal{F}_{2,2,d} $ is regular everywhere.
By Theorem \ref{thm:cV at most EDdegree},
$$
 {\rm EDdegree}(\mathcal{F}_{2,2,d}) \geq C^\#(\mathcal{F}_{2,2,d}) = 8,
$$
for $d\geq 4$.
From {\tt Macaulay2} \cite{M2} computations,  ${\rm EDdegree}(\mathcal{F}_{2,2,d})$ equals $16,34,64,98$ when $d=4,6,8,10$ respectively. This suggests the gap between $C^\#(\mathcal{F}_{2,2,d} )$ and  ${\rm EDdegree}(\mathcal{F}_{2,2,d})$
increases with $d$.
}
\end{example}

\begin{example}
{\rm
Let $\mathcal{E}$ be the  essential variety from \eqref{eqn:essential}.
It is an irreducible variety of codimension three in $\ensuremath{\mathbb{R}}^{3\times 3}$
\cite[Proposition 3.6]{demazure1988}.
By the transfer of smoothness (see the paragraph after the proof of Proposition~\ref{prop:proj transfer}),
we know $\mathcal{E}^\ast = \mathcal{E}\setminus\{0\}$.
Moreover, a straightforward computation verifies that any non-zero essential matrix is a regular point of ${\mathcal E}$.
Thus $\mathcal{E}^{\rm reg} = \mathcal{E}^\ast = \mathcal{E}\setminus\{0\}$, and by Theorem~ \ref{thm:cV at most EDdegree},
we have
\begin{equation} \label{eq:eddegree e variety}
{\rm EDdegree}(\mathcal{E}) \geq  C^\#(\mathcal{E})=6.
\end{equation}
With completely different tools, but entirely motivated by this work, one can prove the equality
${\rm EDdegree}(\mathcal{E})=6$. The details will be described in a forthcoming paper.
}
\end{example}

redpajama_set_name: RedPajamaArXiv


Face brightening toner! Reduces the size of pores, brightens face, reduces inflammation, and helps with acne!
For oily skin add 2 tablespoons of isopropyl alcohol.
Begin by mixing all the ingredients in a jar, cleanse skin twice with a facial cleanser, then apply toner with a cotton pad, sweep upward and outward over entire face and neck, then apply moisturizer and give yourself a face massage. Repeat this process twice a day for better results, such as deeper cleansing, smaller pores and bright, radiant skin, first in the morning and then before you go to bed.


text:
Robert Olen Butler Jr Biography
Nationality: American. Born: Granite City, Illinois, 1945. Education: Northwestern University, B.S. 1967; University of Iowa, M.A. 1969; postgraduate study at the New School for Social Research, 1979-81. Military Service: Military intelligence, U.S. Army, 1969-72: served in Vietnam, became sergeant. Career: Reporter/editor, Electronic News, New York, 1972-73; high school teacher, Granite City, Illinois, 1973-74; reporter, Chicago, 1974-75; editor-in-chief, Energy User News, New York, 1975-85; assistant professor, McNeese State University, Lake Charles, Louisiana, 1985-93, professor of fiction writing, 1993—. Awards: TuDo Chinh Kien award for Outstanding Contributions to American Culture by a Vietnam Vet (Vietnam Veterans of America), 1987; Emily Clark Balch award, 1990; Pulitzer prize, 1993; Richard and Hilda Rosenthal Foundation award (American Academy of Arts and Letters), 1993; Notable Book award (American Library Association), 1993; Guggenheim fellow, 1993; National Endowment for the Arts fellow, 1994. L.H.D., McNeese State University, 1994. Agent: Kim Witherspoon, Witherspoon and Chernoff.
The Alleys of Eden. New York, Horizon Press, 1981.
Sun Dogs. New York, Horizon Press, 1982.
Countrymen of Bones. New York, Horizon Press, 1983.
On Distant Ground. New York, Knopf, 1985.
Wabash. New York, Holt, 1987.
The Deuce. New York, Holt, 1989.
They Whisper. New York, Holt, 1994.
The Deep Green Sea. New York, Holt, 1997.
Mr. Spaceman. New York, Grove Press, 2000.
A Good Scent from a Strange Mountain: Stories. New York, VikingPenguin, 1992.
Tabloid Dreams. New York, Holt, 1996.
Coffee, Cigarettes and A Run in the Park. Decantur, Georgia, Wisteria Press, 1996.
Introduction, Vietnam War Literature: A Catalog by Ken Lopez, Francine Ness, and Tom Congalton. Hadley, Massachusetts, K. Lopez, 1990.
Foreword, Fragments by Jack Fuller. Chicago, University of ChicagoPress, 1997.
Critical Studies:
Conversations with American Novelists: The Best Interviews from the Missouri Review and the American Audio Prose Library, edited by Kay Bonetti, et al, Columbia, University of Missouri Press, 1997.
After writing six novels that received little attention, Robert Olen Butler won the Pulitzer Prize with his 1992 collection of short stories, A Good Scent from a Strange Mountain. What captured the attention of the Pulitzer judges—and of Butler's rapidly growing audience—is that these short stories are all narrated in first person by Vietnamese characters—businessmen, housewives, war veterans, all immigrants to the United States. This is a remarkable and unique achievement when compared with the "Vietnam fiction" of other authors such as Tim O'Brien and Elizabeth Scarborough. Butler speaks for the Vietnamese themselves, rather than about Americans affected by the war.
Butler's father taught theater in Illinois, so he earned a B.A. and M.A. in playwriting during the 1960s, assuming he would become an actor or dramatist. But after graduating in 1969, he anticipated conscription into military duty in Vietnam. He signed up for a position in counterintelligence, reckoning he could stay Stateside. He spent a year learning to speak Vietnamese fluently and in 1971 lived in Bien Ha and Saigon.
Returning to the States, Butler married, divorced, worked at journals and newspapers, remarried, and realized his calling was in fiction, not drama. He began writing novels on the commuter train from Long Island to Manhattan. When he flew to southern Louisiana in 1985 to accept a position as creative writing instructor at McNeese State University, the view from the plane of the wetlands and rice paddies reminded him so much of Vietnam that he moved there at once, and was please to find a thriving community of Vietnamese expatriates.
Because of these events, much of Butler's fiction is concerned with the search for a new home (or a lost one), the sadness of families separated by war and death, relationships between mothers and daughters, fathers and sons.
In Butler's first book, The Alleys of Eden, Clifford Wilkes, an American deserter in Vietnam, shares with lovely Lanh a blissful, passionate affair. The night the Viet Cong surround Saigon, Cliff must decide whether to run for the airlift, whether to bring Lanh with him, or whether to die now rather than risk capture and torture at the hands of the communists. During this night, which comprises the first half of the book, Cliff relives scenes from his past involving separation and loss: the death of his father, the disaffection of his mother, the flight of his best friend to Canada, his divorce. Cliff determines not to abandon Lanh, and they escape Saigon together. However, they cannot restore their happiness in America. With a culture shock too great, the two lovers drift into loneliness toward a hopeless conclusion.
When Cliff recalls his stint as a general infantryman in Alleys, Butler depicts the event that led him to desert. Two of his compatriots in the army were Wilson Hand and Captain David Fleming, the former of whom had been a prisoner of the Viet Cong and the latter his daring rescuer. Their stories comprise the next two novels, Sun Dogs and On Distant Ground.
In Sun Dogs, Wilson works for Royal Petroleum, which sends him from its New York office to Alaska to investigate theft of information. Wilson detests the city, where he feels "dead and buried" in its "clutter." He yearns to stay in the Alaskan outback, with its purity of landscape and stark living conditions. However, like the "sun dogs" Wilson sees in the Alaskan sky—illusory suns formed by atmospheric ice crystals—two horrors dog him: the suicide of his ex-wife and the torture of imprisonment by the Viet Cong. Wilson has an affair with the beautiful, untrustworthy Marta, and becomes friends for life with Clyde, a loyal bush pilot. Butler peels away lies and red herrings, webs of deceit and murderous schemes, to reveal not merely a mystery about corporate greed and reserves of fuel, but an unexpected drama about the fuels that keep the body and spirit alive.
In On Distant Ground, David Fleming is court-martialed for having "aided the enemy." An emotionally aloof man, he had been strangely affected by spying a graffito on a South Vietnamese prison wall: "Hygiene is healthful." The irony and courage of these words speak to him as those written by a kindred soul. Finding the prisoner incarcerated there, a Viet Cong officer named Tuyen, whose name suggests "twin," became an obsession. When at last he tracked Tuyen to a penal colony, he hijacked a helicopter and freed the prisoner. Years later, he is still unable to articulate even to his wife why he did this, and the jury sentences him to a dishonorable discharge.
Even as David and his wife see their first son born, circumstantial evidence convinces David that he also had a son by a Vietnamese woman while serving there. Remembering his own unkind father, he determines to find the boy. With his wife's reluctant compliance, he returns to Vietnam—during the very week that the communists overrun Saigon. While dodging various enemies, he learns the fate of his lover and finds a boy who may be their son. His desperate attempt to get them both out of the country alive culminates in a startling re-encounter with Tuyen, who tells David a truth that forces all of his experiences into new perspective.
Countrymen of Bones chronicles the rivalry of an archaeologist, Darrell Reeves, and a military physicist, Lloyd Coulter, against the backdrop of New Mexico on the eve of the first detonation of an atomic weapon at Los Alamos. The subplots involve the Manhattan Project, run by J. Robert Oppenheimer, who provides the theme of the novel with his famous, awestruck, grieving remark: "I am become Death, the shatterer of worlds"; and a love affair between Reeves and Army private Anna Brown. Human failings such as opportunism and unrestrained anger become the shatterers of private worlds in this work.
At his new home in Louisiana, Butler drew upon autobiographical detail again for Wabash. Raised in Granite City, Illinois, Butler had as a teenager worked at a steel plant; his mother had lived there during the Depression years. This novel combines conflicts between the captains of profit (personified by the owner of the Wabash Steel foundry) and the budding communists, between Deborah Cole and her inarticulate, impotent husband Jeremy, and between Deborah's extended family of mother, aunts, and grandmother. As in so many of Butler's novels, the various conflicts parallel and play off each other in ingenious ways.
The Deuce marked a narrative breakthrough for Butler when he chose a first-person voice. The narrator is a teenager who seeks to understand his culture and identity as the son of a South Vietnamese bar girl and of a Vietnam vet who brings him to New Jersey. He used the technique again in A Good Scent from a Strange Mountain.
The fifteen stories therein are told by a war-time translator; an ex-spy in South Vietnam who goes to grotesque, Roald Dahl-esque measures to keep his gorgeous wife sexually faithful; a mother speaking to her unborn child; by Catholics and Buddhists; by the wealthy and the desperate; and in the title story by a saddened, dying old man who hallucinates a visitation by Ho Chi Minh. Some of these stories are slight, with a single, poignant point to make about the human condition. Most work on many levels to gather such themes as love, envy, loneliness, sensuality, miscommunication, hatred, vengefulness, and spiritual redemption into tales about ghosts, reincarnation, and cultural assimilation.
A Good Scent thrust Butler into the international spotlight. Besides the Pulitzer, it earned the Rosenthal Foundation Award from the American Academy of Arts and Letters, the Southern Review / Louisiana State University Prize for Short Fiction, a nomination for the PEN/Faulkner Award, and fellowships from the Guggenheim Foundation and the National Endowment for the Arts. Butler's earlier works were reissued, and he was commissioned by the Ixtlan film production company to write a screenplay for the collection. The literary world that had barely noticed Butler could now hardly get enough of him.
Butler next wrote a daring novel about sensuality and sexual relationships, They Whisper. The narrative technique is compelling and marvelous, consisting of a stream-of-consciousness recollection-cum-meditation by thirty-five-year-old Ira Holloway on the beauty and desirability of the women he has known, married, and lusted for. Imagining their thoughts and lives in his contemplation of and paean to the wonders of women, Ira seeks to solve the mystery of sexual yearning.
Two more collections followed: Tabloid Dreams, in which Butler takes inspiration from tabloid headlines, and Coffee, Cigarettes, and A Run in the Park, composed of three Vietnam stories. Butler's next foray into the stranger-in-a-strange-land mode came with Mr. Spaceman, superficially a science fiction novel. The stories of Desi, the humanoid extraterrestrial of the title, and of the twelve people who spend the last week of the twentieth century with him are explicitly modeled on the myths of the New Testament. Desi has been charged by his own race to reveal to humanity the existence of other sentient beings in the universe, and struggles with reluctance and fear when he is taken for the returning Messiah. Butler meditates upon language and its limitations, often to very funny effect, as Desi struggles in his precise and literal way to understand humans.
Butler has said in many interviews that he believes fiction can reveal human experience as universally understandable, that its power lies in connecting readers of all cultures through sensuality and emotion.
—Fiona Kelleghan
Octavia Estelle Butler Biography

redpajama_set_name: RedPajamaCommonCrawl


text:
Home Conservative Justice Advocates Orgies
Conservative Justice Advocates Orgies
TBC Staff
(Oliver Burkeman in New York, Friday October 1, 2004, The Guardian)
He is the conservative bastion of the US supreme court, a favourite of
President Bush, and a hunting partner of the vice-president. He has argued vociferously against abortion rights, and in favour of anti-sodomy laws.
But it turns out that there is another side to Justice Antonin Scalia: he
thinks Americans ought to be having more orgies.
Challenged about his views on sexual morality, Justice Scalia surprised his audience at Harvard University, telling them: "I even take the position that sexual orgies eliminate social tensions and ought to be encouraged."
It seems unlikely that this is what President Bush meant when he promised to appoint more judges like Scalia to the court, should the opportunity arise.
Crucially, Justice Scalia is one of the judges in favour of overturning Roe v Wade, the landmark judgment protecting abortion as a constitutional right.
He offered no clue to the logic behind his claim that orgies eliminate
social tensions.
[TBC: This know also, that in the last days perilous times shall come. For men shall be lovers of their own selves, covetous, boasters, proud, blasphemers, disobedient to parents, unthankful, unholy, Without natural affection, trucebreakers, false accusers, incontinent, fierce, despisers of those that are good, Traitors, heady, highminded, lovers of pleasures more than lovers of God; Having a form of godliness, but denying the power thereof: from such turn away (2 Timothy:3:1-5 [1] This know also, that in the last days perilous times shall come. [2] For men shall be lovers of their own selves, covetous, boasters, proud, blasphemers, disobedient to parents, unthankful, unholy, [3] Without natural affection, trucebreakers, false accusers, incontinent, fierce, despisers of those that are good, [4] Traitors, heady, highminded, lovers of pleasures more than lovers of God; [5] Having a form of godliness, but denying the power thereof: from such turn away.
See All...).]
Click here to see our updated Privacy Policy
There are quite a few options for keeping in touch with us:
Subscribe to our email updates here, and choose if you want daily, weekly, or monthly updates.
The Berean Call Mobile App
Most of our content is published in our app for iOS, Andoid, Windows, and Kindle. Click here to get the app!
https://www.facebook.com/thebereancall/
https://twitter.com/thebereancall
https://www.instagram.com/thebereancall/
https://www.youtube.com/user/TheBereanCall/videos
Livestream:
https://livestream.com/bereancallconference
https://vimeo.com/thebereancall
The Berean Call is available by postal mail each month for free. We are willing to send the newsletter to you by regular mail but encourage you to read it online to save printing and mailing costs. Send us a message to add or change your paper mail address.
The Berean Call is also available in a large-print edition at a nominal cost.
If you are outside the U.S. The Berean Call is available by paid subscription .


This plein air study captures a moment in the life of California's migrant farm workers. When taking breaks from harvesting crops, field laborers develop a sense of community by getting together to share meals, play cards, and simply talk.
A Collector's Feast, American Legacy Fine Arts, November 16 – December 15, 2018 CAC 107th Annual Gold Medal Exhibition, Natural History Museum of Los Angeles County, June-July, 2018.


text:
The 2006 Ordina Open was the 2006 edition of the Ordina Open tennis tournament. It was the 17th edition of the tournament and was played on outdoor grass courts in Rosmalen, 's-Hertogenbosch Netherlands. The men's and women's tournament was held from 18 June until 24 June 2006. Mario Ančić and Michaëlla Krajicek won the singles titles.

Finals

Men's singles

 Mario Ančić defeated  Jan Hernych 6–0, 5–7, 7–5

Women's singles

 Michaëlla Krajicek defeated  Dinara Safina 6–3, 6–4

Men's doubles

 Martin Damm /  Leander Paes defeated  Arnaud Clément /  Chris Haggard 6–1, 7–6

Women's doubles

 Yan Zi /  Zheng Jie defeated  Ana Ivanovic /  Maria Kirilenko 3–6, 6–2, 6–2

References

External links

 Men's Singles draw
 Men's Doubles draw
 Men's Qualifying Singles draw

Ordina Open
Ordina Open
Rosmalen Grass Court Championships
2006 in Dutch tennis

redpajama_set_name: RedPajamaWikipedia


text:
Q: Which phrase does not match the others about ratios? Which ratio does not belong w/the other three?

2 parts to 5 parts
2 out of every 5
2 for each 5
2 for every 5

A: 2 out of every 5 is different. There are 5 parts total. 2 parts of something and 3 parts of something else.
The other three have 7 parts total. 2 parts of something and 5 parts of something else.

redpajama_set_name: RedPajamaStackExchange


OilQuickUSA's live demonstration at Demolition Austin created quite the buzz and showed the industry what all the excitement surrounding the OilQuick quick coupling system is all about. Onlookers were impressed with how the coupler allowed operators to change from one attachment to another in seconds without leaving their cab. OilQuick partnered with Kobelco USA, which showcased the system on its 350 high-reach demolition machine. The company also partnered with Bottom Line Equipment and JR Ramon, which both have multiple systems in their fleets. The cost savings of operating with the OilQuick system shouldn't be overshadowed by the increased safety it provides to employees while changing attachments. The OilQuick fully automatic quick coupler system is fast becoming the new standard for demolition, utility and construction contractors.


text:
Die Demirköprü-Talsperre () liegt am Fluss Gediz etwa 120 km östlich der Stadt İzmir in der westtürkischen Provinz Manisa.

Die Talsperre wurde in den Jahren 1954–1960 zur Energiegewinnung, Bewässerung und Abflussregulierung errichtet.
Der  Steinschüttdamm hat eine Höhe von 74 m und besitzt ein Volumen von 4,3 Mio. m³.
Der zugehörige Stausee besitzt eine Wasserfläche von 45,7 km² und ein Speichervolumen von 1022,3 Mio. m³.
Die Talsperre dient der Bewässerung einer Fläche von 99.220 ha.
Das Wasserkraftwerk der Demirköprü-Talsperre verfügt über drei Einheiten zu je 23 Megawatt. Das Regelarbeitsvermögen liegt bei 193 GWh im Jahr.

Einzelnachweise

Wasserkraftwerk in der Türkei
Stausee in Asien
Staudamm in der Türkei
Stausee in der Türkei
SDemirkoprutalsperre
Erbaut in den 1960er Jahren
Steinschüttdamm
Gewässer in der Provinz Manisa
Bauwerk in der Provinz Manisa


text:
Der Fundplatz am Mount Sandel () ist der älteste Beleg für die Ansiedlung mesolithischer Jäger und Sammler auf der irischen Insel. Mount Sandel liegt östlich der Reste eines Forts im Stadtgebiet von Coleraine, im County Londonderry in Nordirland. Der Fundplatz liegt etwa acht Kilometer (Luftlinie) von der Mündung des River Bann in den Atlantik auf einer exponierten Anhöhe.

Beschreibung
Die im 19. Jahrhundert entdeckte und zwischen 1973 und 1977 von Peter Woodman ausgegrabene 700 m² große archäologische Stätte, ist laut Radiokohlenstoffdatierung 9000 Jahre alt und von größter Bedeutung für die Vorgeschichte Westeuropas.

Der Flurname Mount Sandel bezeichnet auch eine eisenzeitliche Befestigungsanlage. Einige Historiker lokalisieren an diesem Platz zudem den mittelalterlichen Ort Kill Santain oder Kilsandel. Dieser war Lagerplatz und befestigte Stützpunkt von John de Courcy während der anglonormannischen Eroberung im 12. Jahrhundert.

Woodman fand Belege für sechs Rundhütten von etwa sechs Metern Durchmesser mit einem zentralen Herd. Die siebte Hütte ist kleiner, nur etwa drei Meter im Durchmesser und hat einen äußeren Herd. Die Hütten wurden aus einem Gerüst aus gebogenen, kreisförmig in Pfostenlöchern in den Boden gesteckten Ästen hergestellt, die wahrscheinlich mit Fellen bedeckt waren.

Eine Vielzahl von Mikrolithen (winzige Werkzeuge aus Stein) wurden gefunden, aber auch Äxte aus Feuerstein, Nadeln, ungleichseitig dreieckige Pfeilspitzen und Messer. Obwohl die Erhaltungsbedingungen nicht gut waren, wurden in einer Feuerstelle Haselnussschalen und Knochenfragmente gefunden. Eine Reihe von Markierungen im Boden wurden als Reste von Gerüsten zum Trocknen von Fischen interpretiert. Reste von Rotwild, Federwild und Wildschweinen sowie von Seehunden, Aalen, Makrelen und Muscheln wurden festgestellt.

Der Platz wurde ganzjährig, gleichzeitig aber von nicht mehr als 15 Menschen bewohnt. Gegen 6000 v. Chr. wurde Mount Sandel verlassen.

Siehe auch
 Jagd- und Sammelbelege aus der irischen Vorzeit

Literatur
 Laurence Flanaghan: Ancient Ireland – Life before the Celts. Gill & Macmillan, 1998, ISBN 0-7171-2433-9; S. 5, 6, 19–26, 28, 143, 144, 149, 152, 153, 166, 184, 200, 201, 208

Einzelnachweise

Weblinks
 Mit Bild der Ausgrabung

Archäologischer Fundplatz in Causeway Coast and Glens
Archäologischer Fundplatz in Europa
County Londonderry
Archäologischer Fundplatz (Mittelsteinzeit)
Vor- und Frühgeschichte Irlands
Archäologischer Fundplatz (Eisenzeit)
Scheduled Monument in Nordirland
Ur- und Frühgeschichte (Westeuropa)


At this point within Final Fantasy XIV it is unlikely we will see another spilt job that has the same treatment Arcanist got but I do feel there is a change they could make to give White Mages and Astrologian a DPS "job" so that they can queue up as a DPS (admit a weak one) or otherwise become a DPS where needed such as fights where only one healer might be needed.
For the rest of this example, I will only use White Mage but the same logic can be applied for Astrologian. In the world of Hydaelyn Geomancers draw that power from the "star below" which is Hydaelyn itself – pretty much the same as Conjurers do.
So my idea is simple. We have four new skills to White Mages. The first of these skills being "Geomancer Stance" or something like that which massively cuts the healing output of White Mage, provides a damage output increase but also allows the usage of three new skills Flood, Toronado, Quake.
While I am not one to be able to balance the job the idea behind these three skills is to give an increase of DPS in order to match the weakest DPS in the game rather than outright buffing White Mages spells and also to allow the developers to add on whatever mechanics they want for the job itself to make it more interesting.
Now you may be wondering what is the downside to this idea? After all, if White Mages could just turn on "Geostance" then they are basically a fifth DPS and we get back to the old debate about Cleric Stance. Well as I said earlier the idea is to allow a White Mage (and Astrologian) to be able to queue up to a duty as a DPS (duty finder could be edited that this can't work for queueing up for certain duties and so on) but also to change in a duty for cases only one healer is needed. This would be done the same way an Astrologian sect works – once you are in "Geostance" and gain agro you are unable to turn this off and this applies the other way as well as you would unable to turn it on during a fight.


text:
Christmas Tree Farms Open In North Carolina
Belews Creek Farm: Business Picked Up Despite Bad Economy
Adam Orr
Friday was opening day for Christmas tree farms across North Carolina, and farm owners said people are already heading out to pick out the perfect tree."Starting this year, we're coming (to the tree farm) for Thanksgiving. And then, (we'll) get the tree today (Friday), go home to decorate the tree and have Christmas with my family tomorrow (Saturday)," customer Beth Pendley said.Picking out a tree together with the entire family is a tradition that Rail Fence Farm owner Kenneth Vanhoy said he loves to see."We have some who come out and take the football and stay for a good while. Some people even bring picnics. It's a fun time of year," Vanhoy said.Vanhoy's farm has been growing Christmas trees for 44 years, and despite family budgets being tight the past few years, Vanhoy said he hasn't seen a dip in business."Then again, more people are thinking local and trying to stay closer, so I think (that's why) our business has really picked up," Vanhoy said.North Carolina is one of the top producers of Christmas trees in the nation, and Vanhoy said his farm will sell about 200 trees in the next month.Prices on real trees range, and at Rail Fence Farm, Vanhoy said it'll cost between $25 and $40 depending on the height and shape.
GREENSBORO, N.C. —
Friday was opening day for Christmas tree farms across North Carolina, and farm owners said people are already heading out to pick out the perfect tree.
"Starting this year, we're coming (to the tree farm) for Thanksgiving. And then, (we'll) get the tree today (Friday), go home to decorate the tree and have Christmas with my family tomorrow (Saturday)," customer Beth Pendley said.
Picking out a tree together with the entire family is a tradition that Rail Fence Farm owner Kenneth Vanhoy said he loves to see.
"We have some who come out and take the football and stay for a good while. Some people even bring picnics. It's a fun time of year," Vanhoy said.
Vanhoy's farm has been growing Christmas trees for 44 years, and despite family budgets being tight the past few years, Vanhoy said he hasn't seen a dip in business.
"Then again, more people are thinking local and trying to stay closer, so I think (that's why) our business has really picked up," Vanhoy said.
North Carolina is one of the top producers of Christmas trees in the nation, and Vanhoy said his farm will sell about 200 trees in the next month.
Prices on real trees range, and at Rail Fence Farm, Vanhoy said it'll cost between $25 and $40 depending on the height and shape.


text:
Q: GUI does not redraw while stepping in Debug Mode In any .NET 2.0 Winforms application, if you run the code in Debug Mode, and you either hit a breakpoint or otherwise Step Into the code line by line, and you want to see the GUI of the application being debugged, then .NET does not draw the application screen.
For example, I have an application which writes some messages to a TextBox on a Form. When I debug the code step by step, or when a breakpoint is hit, I want to see what all messages are logged in the TextBox, but if I press Alt-Tab to switch from VS2005 window to the WinForms application window, all I see is white color. The form is not redrawn, until you press F5 in the debug mode in VS2005.
What is the reason for this, and is there a way to overcome this, without introducing any threads in the code?

A:
What is the reason for this

While you're debugging, you're effectively blocking the UI thread - you're manually stepping through its execution flow. The UI thread can't draw the UI while you're stopping it from executing.

and is there a way to overcome this

You could try calling Application.DoEvents() manually, but I'd generally recommend against it.
It would be better to just wait until you got to the end of the method and let the UI redraw itself normally. If your method is very long (in terms of time) then bear in mind that when not debugging, the UI still wouldn't be able to update itself while that method is executing. This may cause you to change your design (it's hard to tell just from the information we've got at the moment).

A: The reason for this is because you can only have one UI thread, and when you enter your method that updates that code that code begins blocking the UI thread. It will not update until your method exits.
Here is a good SO on message pumps, which are what drive the UI updates
You should be able to use Add Watch/Quick Watch to look at any values at the time of debugging. This sounds like what you are really looking for, anyway.

A: Like everyone else has said in answers and comments, the UI thread is blocked so it cannot be redrawn.
However, if all you want to do is see the GUI, and not interact with it, and you are running Windows 7/8 (which sounds unlikely since you're using VS2005) and haven't disabled aero peek, you can mouse over your application in the task bar and Windows will show the preview thumbnail. When you mouse over the thumbnail, you can "peek" at the application even when the breakpoint is blocking the UI thread.


The elevated position of the beautiful Villa Nefeli takes full advantage of the most amazing landscape towards the old and new villages of Skopelos, one can't help but marvel at the sight before you. As you enter the spacious open plan interior of this imposing villa, it is evident that much thought and care has been taken in its design. The whole house is cool and comfortable; calming colour schemes combined with traditional features are a gentle reminder that you are in a unique part of the Aegean.
The fully equipped kitchen joins the dining area but it is without doubt that most of your meals will be enjoyed outdoors on the shaded terrace. There is a double bedroom on the ground floor with French doors leading out onto the sunny patio and pool area. This is also where the main bathroom facilities are located. From the ground floor, stairs lead up to a further two bedrooms, one double and one twin. French doors take you out onto a shared verandah with the most spectacular panoramic views of this most beautiful part of the Sporades and out over the stunning Aegean sea.
Outside the view is undoubtedly the dominant feature and it is with this in mind that the gardens have been landscaped and terraced to not only blend in with the verdant surroundings but to maximise your enjoyment of the area. The pool enjoys magnificent views out over rural villages with majestic mountains in the distance. The most popular beach is Stafylos, which is approximately 15 minutes away by car. Around the hillsides are many donkey trails, fabulous for exploring the local nature and enjoying a gentle ramble in the most beautiful setting. This villa is certainly ideal for families and groups but equally suitable for a lovely retreat so much so you may not want to leave!
NB: Villa Nefeli is reached via a long unmade rough track.