text:
\begin{document}
\date{8 June, 2004.}
\title{Lower bound for energies of harmonic tangent unit-vector fields on convex
polyhedra}
\author{A Majumdar$^{\dag,\ddag}$,
JM Robbins$^{\dag}$ \& M Zyskin$^{\dag}$\thanks{  {\tt
  m.zyskin@bristol.ac.uk}}\\
$^\dag$School of Mathematics\\ University of Bristol, University Walk,
Bristol BS8 1TW, UK\\
$^\ddag$  Hewlett-Packard Laboratories\\
 Filton Road, Stoke Gifford, Bristol BS12 6QZ, UK}
\maketitle

\noindent  {\bf Abstract.} We derive a lower bound for energies of harmonic maps of convex
  polyhedra in $ \Rr^3 $ to the unit sphere $S^2,$ with tangent
  boundary conditions on the faces.  We also establish that $C^\infty$
  maps, satisfying tangent boundary conditions, are dense with respect
  to the Sobolev norm, in the space of continuous tangent maps of
  finite energy.

\noindent  {\bf Mathematics Subject Classifications (2000).} 58E20, 35J55.

\noindent  {\bf Key Words.} Bi-stable nematic devices, liquid crystals, harmonic maps, polyhedra, tangent boundary conditions, lower bounds.

\newpage

\tiny
\hspace*{70mm} \mbox{"The unburdened mind is joy supreme"}
\\
\hspace*{80mm} \mbox{Gel-sang-gya-tso, 1708-1757.}
\normalsize

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

We consider maps $\nvec$ from a convex polyhedron $P \subset \Rr^3 $
to the unit sphere $S^2$, which we regard as unit-vector fields on
$P$.  $\nvec$ is said to satisfy {\em tangent boundary conditions}, or
to be {\em tangent}, if, on the faces of $P$, $\nvec$ is tangent to
the faces.  This implies that, on the edges of $P$, $\nvec$ is
parallel to the edges, and, therefore, discontinuous at the vertices.
We consider maps which are continuous away from vertices and which
belong to the Sobolev space $W^{1,2}(P)$. There are homotopically
inequivalent classes of such maps \cite{rz}.  In this paper, we derive
a lower bound for the energy
\begin{equation}\label{eq: one-constant}
E[\nvec] = \int\int\int_P \lb \nabla \nvec\rb^2 dV
\end{equation}
for each homotopy class.  We also establish that $C^\infty$ tangent
unit-vector fields are dense in the space of continuous tangent
unit-vector fields in $W^{1,2}(P)$ with respect to the Sobolev norm.

This work is part of a study of liquid crystals in polyhedral
geometries started in \cite{rz}.  We have been motivated by
applications to the design of bi-stable liquid crystal displays (see,
eg, \cite{newtonspiller}) as well as by mathematical considerations.

A nematic liquid crystal is a suspension
of rod-shaped molecules in a liquid substrate.  The molecules have a
preferred average orientation at every point in space.  This preferred
orientation is described by a {\it director field} -- a unit-vector
field with opposite orientations identified

\cite{dg}.  We are
only considering continuous director fields in a simply connected
domain, in which case an orientation can be chosen arbitrarily at one
point and defined elsewhere by continuity, thus yielding a unit-vector
field $\nvec$.  (Quasi-) stable configurations are (local) minima of a
certain energy functional, the Frank energy \cite{dg}
\begin{align}
  \label{eq:Frank}
  E_F[\nvec] = \int\int\int_V &\big( K_1 (\text{div}\, \nvec)^2 + K_2 (\nvec \cdot
  \text{curl}\, \nvec)^2 + K_3 (\nvec \times \text{curl}\, \nvec)^2 +\notag\\
&\quad   + K_4 \,\text{div}\,  \ls ({\mathbf{n}}\cdot \nabla) {\mathbf{n}} -
{\mathbf{n}}\, \text{div}\, {\mathbf{n}} \rs \big)
  \,dV.
\end{align}
In the  `one-constant approximation' $K_1=K_2=K_3=K_4=1$, and the Frank energy
reduces to \eqref{eq: one-constant}.

Solutions of the Euler-Lagrange equation corresponding to \eqref{eq:
  one-constant}, subject to appropriate boundary conditions, are
harmonic maps of $P$ to $S^2$.  The boundary conditions are determined
by the substrate and surface treatment used.  In the cases being
considered here, it is strongly energetically favorable for the vector
field to be tangent to the boundary.

Let $\bar{P} \subset \Rr^3$ be a convex

polyhedron.
Let $\vvec^a$, $a = 1,
\ldots, v$, label the vertices, $E^b$, $b = 1,\ldots, e$, the edges
and $F^c$, $c = 1,\ldots, f$, the faces of $P$ respectively. $ \Fvec^c$
is the outward unit-normal vector to $F^c.$ Let $P $ denote
$\bar{P}$ without its vertices.

Let $\czero(P)
$ denote the space of continuous tangent unit-vector fields on $P$.
We shall also have occasion to refer to
$\ck(P)$, the space of $C^k$ tangent unit-vector fields, and
$\cinf(P)$, the space of smooth tangent unit-vector fields.

We say that   $\nvec$, $\npvec \in
\czero(P)$ are  homotopic,  $\nvec \sim \npvec$, if there exists
a continuous map $\Hvec: P\times [0,1]\rightarrow S^2;
(\xvec,t)\mapsto \Hvec_t(\xvec)$, such that $\Hvec_t \in\czero(P)$ for
$t \in [0,1]$
   and
$\Hvec_0 = \nvec$, $\Hvec_1 = \npvec$.

It is shown in \cite{rz} that homotopy classes of tangent unit-vector
fields are classified by a set of invariants, which we call
{\it edge orientations}, {\it kink numbers} and {\it wrapping
  numbers}.  These are defined as follows:\\
\noindent {\em Edge orientations:} The edge orientation $\evec^b(\nvec)$ is the
value of $\nvec$ on the edge $E^b.$

\vspace{1mm}
\noindent {\em Kink numbers:}

Let $\ga^{ac}$ denote a path on $F^c$, positively oriented with
respect to $\Fvec^c$, between the pair of edges with common
vertex $\vvec^a$.  The image of $\ga^{ac}$ under $\nvec$ describes an
arc on $C^c$, the great circle in $S^2$ parallel to $F^c$.  The kink
number $k^{ac}(\nvec)$ is the degree of the closed path on $C^c$
obtained by closing $\nvec(\ga^{ac})$ with the shortest arc between
its endpoints.  Since $\nvec$ is continuous away from vertices, its restriction
to any closed path on $F^c$ away from vertices has degree
zero.  This implies the following sum rule
for the kink numbers: Let
$q^c(\nvec)$ denote the number of vertices of $F^c$ at which the edge
orientations are oppositely oriented with respect to the normal
$\Fvec^c$.  Then $\sum_{\vvec^a \in F^c} k^{ac}(\nvec) = 1- \half q^c(\nvec)$.

\vspace{1mm}

\noindent {\em Wrapping numbers.} Choose $\svec\in S^2$ such that
$\svec$ is not tangent to any of the faces of $P$.

For each vertex $\vvec^a$, choose an outward-oriented surface
$S^a\subset P$ which separates $\vvec^a$ from the other vertices.  The
boundary of $S^a$ lies on those faces of $P$ which meet at $\vvec^a$.

Construct a new map $\nuvec^a: S^a \rightarrow S^2$ which coincides
with $\nvec$ on $\partial S^a$ and whose image does not contain $\svec$.

The wrapping number $w^a(\nvec)$ is the degree of the map
$S^2\rightarrow S^2$ obtained by gluing the maps $\nvec|_{S^a}$ and
$\nuvec^a$ along the boundary of $S^a$.
The fact that $\nvec$ is continuous on
$P$ implies that $\sum_{a=1}^v w^a(\nvec) = 0$.

In what follows, we denote the invariants collectively by $\inv =
\{\evec^b, k^{ac}, w^a\}$.

The paper is organized as follows.
A lower bound for the energies of
$C^\infty$ tangent unit-vector fields in terms of the invariants is
given in Theorem~\ref{th: th1}.  The derivation adapts methods of
\cite{bl} to the tangent boundary-value problem treated here.
(Boundary value problems were not considered in \cite{bl}. Also, the
lower bound involves not only the degree of the map on two-dimensional
surfaces surrounding the vertices, but also kink numbers and edge
orientations).  In Theorem~\ref{thm: thm 2} we establish that
$C^\infty $ tangent unit-vector fields are dense in the space of
continuous Sobolev tangent unit-vector fields with respect to the
Sobolev norm.  The proof requires certain smoothings of the vector
fields which lie outside the scope of the standard Meyers-Serrin
theorem \cite{adams}.  Thus, the
lower bound of Theorem~\ref{th: th1} extends to $\czero(P)\cap
W^{1,2}(P)$.
Section~\ref{sec:discussion} contains a discussion of the results.

\section{Lower bounds for energies of harmonic maps}\label{sec: energies}
\begin{defn}\label{def:energy}
  The minimal energy $M(h)$ of maps in homotopy class $h$ is defined by
\begin{equation}
M(h) = \inf_{ \genfrac{}{}{0pt}{}{\nvec \in \czero (P) \bigcap
      W^{1,2} (P),}{ \text{ inv} (\nvec) = h}} E [ \nvec],
\label{energyMinimimal}
\end{equation}
where $W^{1,2}(P)$ is the Sobolev space
\begin{equation}\label{eq: Sobolev}
W^{1,2}(P)= \lbr \nvec\left\vert \rd
{\nabla }\nvec \in L^2(P) \rbr
\end{equation}
and
\bge
E[\nvec] =  \int\int\int_P (\nabla \nvec)^2\,dV=
\dst\int\dst\int\dst\int_P  \partial_a n_b \partial_a n_b
\,dV=
\|
\nvec\|^2_{W^{1,2}(P)}.
\label{energy}
\ene
\end{defn}

If the infimum is actually achieved by some tangent unit-vector
field $\nvec \in \ctwo(P), $ then $\nvec$ satisfies the Euler-Lagrange
equation \bge \Delta \nvec - \lt \nvec, \Delta \nvec \rt \nvec =0,
\ene with boundary conditions $\ld \nvec \cdot {\Fvec}^c \rv_{F^c}
=0,$ $((\Fvec^c\cdot\ld \nabla)\nvec) \cdot \lb \nvec \times
\Fvec^c\rb\rv_{F^c} =0$ on the faces of $P.$

It will be shown in Section~\ref{sec: approximation} that $\cinf
(P)\bigcap W^{1,2} (P) $ is dense in $\czero (P)
\bigcap W^{1,2} (P)$ with respect to the Sobolev norm.  Thus, to
compute $M(h)$, it suffices to consider smooth maps only.

It is straightforward to show (a demonstration is given in \cite{bl}) that
the energy density $\rho = (\nabla \nvec)^2$ satisfies the inequality

\begin{equation}

 \rho \geq
2 \left|\nvec^* \omega \right|,
\label{e>d}

\end{equation}
where $\omega$ is the area-form on $S^2$, normalized to have
area $4\pi$, and $|\nvec^*\omega|$ is the Euclidean norm of its
pull-back.  Indeed, since $(\nvec\cdot\nabla)\nvec = 0$,
we may write
\begin{equation}\label{eq: alphas}
 \nabla
\nvec = \alpha_1 {\boldsymbol \tau}_1 + \alpha_2 {\boldsymbol \tau}_2,
\end{equation}
where $ {\boldsymbol\tau}_1$, ${\boldsymbol \tau}_2$ constitute a
(locally defined) orthonormal basis for $T_{\nvec}S^2$, and
$\alpha_1$, $\alpha_2$ are (locally defined) one-forms on $P$. It
follows that $\rho = |\al_1|^2 + |\al_2|^2$, while $|\nvec^* \omega |
= \left|\al_1\wedge \al_2\right|^2 = |\al_1|^2 |\al_2|^2
-(\al_1\cdot\al_2)^2$, where $\al_1\cdot\al_2$ denotes the Euclidean
inner-product on forms.  Therefore, $\rho^2- 4 \left| \nvec^* \omega
\right|^2 = (|\al_1|^2 -|\al_2|^2)^2 + 4 (\al_1\cdot\al_2)^2 \geq 0.$

\begin{rem}
Suppose $\alpha_1\wedge\alpha_2 \ne 0$.
From (\ref{eq: alphas}), if $\iota_X (\alpha_1\wedge\alpha_2) = 0$
for some vector field $X$,
then
$\iota_X \nabla \nvec = 0$; ie, $\nvec$ is constant on
the characteristics of $\al_1\wedge \al_2.$
\end{rem}
\begin{rem}
  If we have equality in (\ref{e>d}), then $|\al_1|^2 =|\al_2|^2$ and
  $\al_1\cdot\al_2=0,$ so that $\nabla \nvec$, regarded as a map from
  the orthgonal complement of the characteristic distribution of
  $\al_1\wedge \al_2$ to $T_\nvec S^2$, is conformal.
\end{rem}

\begin{defn}
The {\it trapped area}  $\Omega^a(\nvec)$ at a vertex $\vvec^a$ is the
area
(as a proportion of the area of $S^2$)
 of the image under $\nvec$ of an outward-oriented surface, $S^a \subset P$,  which
 separates $\vvec^a$ from the other vertices, ie
\begin{equation}\label{eq: Q}
\Omega^a(\nvec) = \frac{1}{4\pi} \int\int_{S^a} \nvec^* \omega.
\end{equation}
\end{defn}

The trapped areas are homotopy
invariants, and may be expressed in terms of edge orientations, kink
numbers and wrapping numbers as follows (see \cite{rz} for details).
Given a vertex $\vvec^a$, let $K^a$ be the geodesic polygon on $S^2$ with
vertices $\evec^{b_1},\ldots \evec^{b_{m}}$ given by the edge
orientations of the edges $E^{b_1},\ldots, E^{b_m}$ which are incident at
$\vvec^a$ (the edges are ordered consecutively with respect to the
outward normal on $S^a$).  Then
\begin{equation}
\Omega^a =
 w^a  -  \half {\sum_{c}}'
\sgn(\Fvec^c\cdot\svec) k^{ac}
+  \sum_{j=2}^{m-1} \left(\frac{1}{4\pi}A(\evec^{b_1},\evec^{b_j},\evec^{b_{j+1}}) -
\sigma(\evec^{b_1},\evec^{b_j},\evec^{b_{j+1}})\right),
\label{Omega[invariants]}
\end{equation}
where the sum $\sum_{c}'$ is taken over the faces $F^c$ incident at
$\vvec^a $, $A(\avec,\bvec,\cvec)\in (-2\pi,2\pi)$ is the oriented
area of the spherical triangle with vertices $\{\avec$, $\bvec$,
$\cvec$\} and $\sigma(\avec,\bvec,\cvec)$ is equal to $\sgn
((\avec\times\bvec)\cdot \svec)$ if $\svec$  is
contained in the spherical triangle with vertices $\{\avec$, $\bvec$,
$\cvec$\} and is zero otherwise.
The trapped areas are typically not integer-valued.  However, they

satisfy the sum rule (related to the sum rule for the wrapping numbers)
\begin{equation}\label{eq: trapped sum}
 \dst\sum_a \Omega^a   = 0
\end{equation}
(this follows from the fact that the map $\nvec:\partial P\rightarrow
S^2$  is contractible).

\begin{thm}
  \label{th: th1}
The minimal energy $M(h)$
is bounded below  by
\begin{equation}
 M(h) \geq \max_{\lbr \xi^a:  \vert \xi^a-\xi^{a'}\vert  \leq
\left|\vvec^a - \vvec^{a'}
\right|
\rbr}
\lb8\pi \dst\sum_a \Omega^a \xi^a\rb > 0.
\label{low_bound}
\end{equation}
\end{thm}
\begin{rem}
The bound is given in terms of a finite-dimensional linear
optimization problem with linear constraints, whose solution can be
found algorithmically using standard methods.
\end{rem}

\begin{proof}
  Let $\nvec \in \cinf(P)\cap W^{1,2}(P)$ with $\inv(\nvec) = h$.  Let
  $\xi \in \Lip_1(P)$, the space of Lipschitz functions on $P$ with
  Lipschitz constant less then one.  Then $\xi$ is almost-everywhere
  differentiable with $\vert d \xi \vert \leq 1$.  It follows from
  \eqref{e>d}
that
\begin{equation}
E[\nvec] =  \int\int\int_{P} \rho \,dV \geq 2  \int\int\int_{P} \left|\nvec^* \omega \right|
dV \geq  2  \int\int\int_{P}    d \xi  \wedge \nvec^* \omega.
\label{eq: xi}
\end{equation}
We remove infinitesimal neighbourhoods of the vertices from the domain
of integration and integrate by parts in the last expression.  As $d\,
\nvec^* \omega = \nvec^* d\, \omega =0$, the volume integral vanishes.
Because $\nvec$ is tangent, $\nvec^* \omega$ vanishes on the faces
$F^c$, so that the only contribution to the surface integral is from
the boundaries of the excised infinitesimal neighbourhoods.  On these
boundaries, $\xi$ can be replaced by its values at the vertices.
Recalling (\ref{eq: Q}), we obtain
\begin{equation}
  \label{eq:identity}
   \int\int\int_{P}  d \xi  \wedge \nvec^* \omega = 8\pi \dst\sum_a \Omega^a
\xi(\vvec^a).
\end{equation}
Since (\ref{eq: xi}) and (\ref{eq:identity}) hold for all $\nvec\in
\cinf(P)\cap W^{1,2}(P)$,
it follows that
 \begin{equation}
 M(h)\geq \sup_{_{\xi \in \Lip_1(P)}}  \lb  8\pi \dst\sum_a \Omega^a
 \xi({
\vvec}^a)
\rb.
\label{xisup}
 \end{equation}

$\xi\in  \Lip_1(P)$ implies the constraints
$\vert \xi(\vvec^a)-\xi(\vvec^{a'})
\vert \leq \vert\vvec^a- \vvec^{a'}\vert$.
Conversely, given a set of
$\xi^a$'s satisfying these constraints, we can construct a function $\xi \in
\Lip_1(P)$ with $\xi(\vvec^a)=\xi^a$, eg by letting
$\xi(\rvec): = \max_{a} \lb \xi^a -  |\rvec - \vvec^a| \rb$.
Therefore,
\begin{equation}
\sup_{\xi \in \Lip_1} \lb 8\pi \dst\sum_a \Omega^a \xi(\vvec^a)  \rb =
\max_{\lbr \xi^a:  \vert \xi^a-\xi^{a'}\vert  \leq \vert \vvec^a -
\vvec^{a'}\vert \rbr}
\lb 8\pi \dst\sum_a \Omega^a \xi^a\rb,
\label{optimise_xi}
\end{equation}
which together with (\ref{xisup}) gives the required lower bound.
Note that \eqref{eq: trapped sum} implies that
the optimal $\xi^a$'s are determined up to an additive constant, which we
can fix, say, by setting $\xi^1=0.$  The constraints ensure
that the feasible set is nonempty and bounded;
thus the maximal value in the
right-hand side is finite. It is obvious that that maximal value is
positive, so that
the lower bound is nontrivial.
\end{proof}
\begin{rem}
  The maximisation problem which appears in the lower bound
  \eqref{low_bound} can be replaced by its equivalent dual
  minimisation problem, as in \cite{bl}.

\end{rem}

\section{Approximating by smooth tangent maps}\label{sec: approximation}
In Theorem \ref{thm: thm 2} below we show that $\cinf(P)\bigcap
W^{1,2} (P) $ is dense in $ \czero (P) \bigcap W^{1,2} (P)$.  This is
accomplished by constructing, for a given continuous tangent
unit-vector field $\nvectilde$, a smooth tangent unit-vector field
$\nvec$ arbitrarily close to $\nvectilde$ with respect to the Sobolev
norm.  Away from the vertices and edges of $P$, $\nvec$ is obtained
from a smooth average of $\nvectilde$ which preserves tangent boundary
conditions.  Neighbourhoods of the vertices and edges require special
treatment, which is dealt with in Lemmas  \ref{lem: vertex extension}
and  \ref{lem: edge}
respectively.

\begin{lem}\label{lem: vertex extension}
  {\bf Vertex extension.}  Let $\vvec^a$ be a vertex of $P.$ Let
  $l^a$ be a ray from $\vvec^a$ into the interior of $P$, and
  introduce local Euclidean coordinates centred at $\vvec^a$ with
  positive $z$-axis along $l^a$.  Let
\begin{equation}\label{eq: Lambda^a}
\Lambda^a(H):=\lbr (x,y,z)\in P
  \,\vert\, 0< z \leq H\rbr
\end{equation}
denote the prism obtained by cutting $P$ by a plane perpendicular to
$l^a$ at a distance $H$ from $\vvec^a$.  Let
\begin{equation}
  \label{eq:Pi}
  \Pi^a = \overline{\Lambda^a(H) \setminus  \Lambda^a(\half H)}
\end{equation}
denote the closed lower half of
$\Lambda^a(H)$.  Given a $C^\infty$ tangent unit-vector field
$\nvectilde$
in $\Pi^a$ and some $\epsilon_3 > 0$ such that
\begin{equation}
E[\nvectilde]:=\dst\int\dst\int\dst\int_{\Pi^a} \lb \nabla
\nvectilde
\rb^2\, dV \leq\epsilon_3,
\label{n_energyPi_small_vert}
\end{equation}
one can construct a $C^\infty$ tangent unit-vector field $ \nvec$ on
$\Lambda^a(H)$ coinciding with $\nvectilde$  on \linebreak
$\Lambda^a(H)\setminus\Lambda^a(\smallfrac{3}{4} H)$
such that
\begin{equation}
E[{\nvec}]:= \dst\int\dst\int\dst\int_{\Lambda^a(H)} \lb \nabla
\nvec\rb^2\,dV \leq
C^a\epsilon_3,
\label{n_energyLa_small_vert}
\end{equation}
where $C^a>0$ is independent of $\nvectilde$, $\epsilon_3$ and $H$.
\end{lem}
\begin{proof}
Let $D := \lbr (u,v) \,\vert \, (Hu,Hv,H) \in P\rbr$ denote the base
of $\Lambda^a(H)$, parameterized by $u$ and $v$.
Introduce new coordinates $(u,v,h)$ by
\bge
\bga
z=Hh, \quad x= Hh u, \quad y =  Hh v ,\quad \text {where}\ 0< h \leq 1, \ \text{and}\ (u,v) \in \mathit{D}.
\ena
\ene
Let $\phivectilde(u,v,h)= \nvectilde( Hh u, Hh v,  Hh).$
From  (\ref{n_energyPi_small_vert}),
\bge
\bga
E[\nvectilde]

=H \dst\int_{\frac{1}{2}}^1  dh\dst\int\dst\int_{D}dudv  \lbr  \lb
\phivectilde_u \rb^2 + \lb  \phivectilde_v \rb^2 + \lb  -u
\phivectilde_u -v \phivectilde_v + h
\phivectilde_h\rb^2
\rbr  \leq \epsilon_3.
\label{n_energyPi_newcoor_vert}
\ena
\ene
Then
\begin{equation} \label{tilde_phi_u vert}
 {\| \phivectilde_u \|}^2_{L^2(\Pi^a)}  \leq
\epsilon_3/H,
\quad {\| \phivectilde_v \|}^2_{L^2(\Pi^a)} \leq
\epsilon_3/H,
\end{equation}
where $ {\|  {\mathbf a}   \|}_{L^2(\Pi^a)}$ means $\lb   \int_{1/2}^1 dh
\int \int_D du dv\,  |{\mathbf a}|^2\rb^{\frac12}$.
Since $u$ and $v$ are bounded on $D$, it follows that
\begin{equation}
\half{\|  \phivectilde_h \|}_{L^2 (\Pi^a)} \leq {\| h
\phivectilde_h \|}_{L^2(\Pi^a)}\leq (\epsilon_3/H)^{\frac12}
+
{\|u \phivectilde_u \|}_{L^2(\Pi^a)} +{\| v \phivectilde_v\|}_{L^2
  (\Pi^a)} \leq
C^a_1
(\epsilon_3/H)^{\frac12} \label{tilde_phi_h vert}
\end{equation}
for some $C^a_1 >0$ independent of $\epsilon_3$, $H$ and
$\nvectilde$.

$\phivectilde$ may be extended to a unit-vector field $\phivec$ on
$\Lambda^a(H)$
according to
\begin{equation}
\phivec(u,v,h)= \phivectilde (u,v,s(h)), \quad 0 < h \leq 1,
\end{equation}
where $s(h)$ is a  $C^\infty$ function on $[0,1]$ with $\half \le s(h)
\le 1$, $s(0) = \half$, $s(h) = h$ for $\frac34\le h \le 1$, and with $s'(h)$ bounded away
from zero.
For  example, we can take
\bge
\bga
s(h)= \frac{3}{4}+ \int_{\frac{3}{4}}^{h}  \al (t) dt,
\\
\al(x) := a+ (1-a) \frac{\int_{0}^{x}  f(t) dt}{\int_{0}^{\frac{3}{4}}  f(t) dt},
\\
f(x): = \lbr \bga \exp\ls \frac{1}{x-\frac{3}{4}}- \frac{1}{x-\frac{1}{2}}\rs, x\in (\frac{1}{2} , \frac{3}{4})
\\
0, \quad  x\not\in (\frac{1}{2} , \frac{3}{4}) \ena\rd,
\\
a := \frac{\ga-\frac{1}{4}}{\ga-\frac{3}{4}}, \ga := \frac{\int_{\frac{1}{2}}^{\frac{3}{4}}  f(t)(\frac{3}{4}-t) dt}{\int_{\frac{1}{2}}^{\frac{3}{4}}  f(t) dt} <\frac{1}{4}.\label{eq: s}
\ena
\ene
We define the corresponding extension of $\nvectilde$ by
\begin{equation}
  \label{eq:n(x,y,z)}
  \nvec(x ,y, z)= \phivec\left(\frac{x}{z},\frac{y}{z},\frac{z}{H}\right),
\quad 0 < z \leq H.
\end{equation}
Clearly, $\nvec$ is a smooth tangent unit-vector field on
$\Lambda^a(H)$.
We estimate its energy (\ref{n_energyLa_small_vert}) as follows.
Let $h(s)$ be the inverse of $s(h).$ Since $h^2(s) s'(h(s))$ and
$1/s'(h(s))$ are bounded on $[0,1]$, say by $C_2$,
\begin{equation}\label{eq: E with elementary inequality}
\bga
E[ {\nvec}] = H \dst\int_{0}^1 dh \dst\int_D du dv  \lbr  \lb
\phivec_u \rb^2 + \lb  \phivec_v \rb^2 + \lb  -u \phivec_u
-v \phivec_v + h \phivec_h\rb^2
\rbr
=
\\
= H \dst\int_{\frac{1}{2}}^1  \frac{ds}{s^\prime (h(s))}\dst\int_{\mathit{D}} du dv  \lbr  \lb
\phivectilde_u \rb^2 + \lb  \phivectilde_v \rb^2 + \lb  -u
\phivectilde_u -v \phivectilde_v +  h(s) s^\prime (h(s))
 \phivectilde_h \rb^2
\rbr \leq
\\
\leq  C_2 H \dst\int_{\frac{1}{2}}^1 dh \dst\int_{\mathit{D}}   du dv \lbr  \lb
\phivectilde_u \rb^2 + \lb  \phivectilde_v \rb^2 + 3 \lb u
\phivectilde_u\rb^2 +3 \lb v \phivectilde_v \rb^2 +3  \lb
 \phivectilde_h \rb^2
\rbr .
\ena
\end{equation}
In the last step of (\ref{eq: E with elementary inequality}),
we have used the elementary inequality $(a + b +
c)^3 \le 3a^3 + 3b^3 + 3c^3$ (similar inqualities are used in what follows).
Using (\ref{tilde_phi_u vert}), \eqref{tilde_phi_h vert}
and the fact that $u$ and $v$ are
bounded on $D$, we obtain
\begin{equation}
  \label{eq:vertex result}
  E[ {\nvec}] \leq C_3^a H\lb {\| \phivectilde_u \|}^2_{L^2(\Pi^a)} +
{\| \phivectilde_v \|}^2_{L^2(\Pi^a)}
+ {\| \phivectilde_h \|}^2_{L^2(\Pi^a)} \rb
\leq C^a \epsilon_3,
\end{equation}
for some $C_3^a > 0$ and $C^a>0$ independent of $\epsilon_3$, $H$ and
$\nvectilde$.
\end{proof}

\begin{lem}\label{lem: edge}{\bf Edge extension.}
  Let $E^b$ be an edge of $P$ between faces $F^+$ and $F^-$.  Let
  $(x,y,z)$ denote local Euclidean coordinates centred about the midpoint of
  $E^b$, with $z$-axis parallel to $E^b$ and positive $y$-axis
  directed into $P$ and lying in the midplane between $F^+$ and $F^-$,
  so that $F^{\pm}$ are given locally by $y \ge 0, x = \pm \tau y $,
where $\tau = \tan\alpha$ and $2\alpha$ is the angle between $F^+$ and $F^-$.
  Let
  \begin{equation}
 \Lambda^b(W,L):=\lbr \label{eq: local edge coords}
  (x,y,z)\,\vert\, 0\leq y \leq W, -\tau y\leq x\leq \tau y,
-\half L \leq z\leq \half L
\rbr
\end{equation}
denote the right prism whose axis has length $L$, is parallel to
$E^b$, and is symmetric about the midpoint of $E^b$, and whose
cross-section is an isosceles triangle with sides in $F^+$ and $F^-$ of
length $ W\sec \alpha$.  Let
\begin{equation}
  \label{eq:Pi^b}
  \Pi^b = \overline{
\Lambda^b(W,L) \setminus \Lambda^b(\half W,L)
}
\end{equation}
denote the closed interior half of this prism.
Let $\evec^b$ be a unit vector parallel to
  $E^b$.  Then, given a $C^\infty$ unit-vector field
$\nvectilde$
on $\Pi^b$
  which satisfies tangent boundary conditions on
$F^\pm$, and constants $\epsilon_1 > 0$ and $0 <
  \epsilon_2 < \half$ such that
\begin{align}
E[\nvectilde]:=\dst\int_{\Pi^b} \lb \nabla \nvectilde\rb^2
&\leq\epsilon_1,
\label{n_energyPi_small}
\\
\max_{\rvec\in \Pi^b} \vert \nvectilde(\rvec) - \evec^b\vert &\leq
\epsilon_2,
\label{n_jumpPi_small}
\end{align}
one can construct a $C^\infty$ unit-vector field $ \nvec$ in
$\Lambda^b(W,L)$
satisfying tangent boundary conditions on $F^\pm$
and coinciding with
$\nvectilde$ on $\Lambda^b(W,L)\setminus\Lambda^b(\smallfrac{3}{4} W,L)$ such that
\bge
E[{\nvec}]:=\dst\int_{\Lambda^b(W,L)} \lb \nabla \nvec\rb^2 \leq C^b
(\epsilon_1 + L
{\epsilon_2}^2),
\label{n_energyLa_small}
\ene
where $C^b$ is independent of $\nvectilde$, $\epsilon_1$,
$\epsilon_2$, $L$ and $W$.
\end{lem}
\begin{proof}
  It is convenient to introduce new  coordinates
$(\xi,\eta,\mu)$ as follows:
\begin{equation}
  \label{eq:edge coords}
  y =  W \eta, \quad  x =  \tau W \eta \xi, \quad z = L\mu .
\end{equation}
Let
\begin{equation}\label{eq: edge phitilde}
\phivectilde(\xi,\eta,\mu)= \nvectilde(\tau W\eta\xi , W\eta , L\mu).
\end{equation}

Then
\begin{equation}
 E[\nvectilde]
= \int_{-\frac12}^{\frac12} d\mu \int_{-1}^1 d\xi \dst\int_{\frac{1}{2}}^1
\frac{d \eta}{\eta}
\lbr
  \frac{L}{\tau}
    (\phivectilde_\xi  )^2
  + L\tau
  (\eta \dst \phivectilde_\eta
     -\xi \dst\phivectilde_\xi
  )^2
  +
  \frac{W^2\tau}{L} \eta^2  ( \dst \phivectilde _\mu
  )^2
\rbr
\leq \epsilon_1.
\label{n_energyPi_newcoor}
\end{equation}

Let $ {\|  \avec   \|}_{L^2(\Pi^b)}$ denote the $L^2$-norm $\lb  \int_{-\frac12}^{\frac12} d\mu
\int_{-1}^1 d\xi \int_{\frac{1}{2}}^1 d\eta\, |\avec|^2
\rb^{\frac{1}{2}}$ of a vector field $\avec$ on $\Pi^b$.  From
(\ref{n_energyPi_newcoor}) one can deduce that
\begin{equation}
{{\|   \phivectilde_\xi  \|}^2_{L^2(\Pi^b)}} \leq
C^b_1\frac{\epsilon_1}{L},\quad
{{\|  \phivectilde_\eta  \|}^2_{L^2(\Pi^b)}}  \leq
C^b_1\frac{\epsilon_1}{L},\quad
{{\| \phivectilde_\mu \|}^2_{L^2(\Pi^b)}}  \leq
C^b_1 \frac{L\epsilon_1}{W^2}
\label{tilde_phi_xi_norm}
\end{equation}
for some $C^b_1 > 0$ independent of $\nvectilde$, $\epsilon_1$,
$\epsilon_2$, $L$ and $W$ (in fact, it suffices to take $C^b_1 =
4(\tau^\frac12 + \tau^{-\frac12})^2$).

We extend $\phivectilde$ to a tangent unit-vector field $\phivec$ on
$\Lambda^b(W,L)$ as follows.   In general, let $a_\parallel = \avec\cdot
\evec^b$ and $\avec_\perp = \avec - a_\parallel \evec^b$ denote the
components of $\avec$ parallel and perpendicular to $\evec^b$.
Let $\psi(\eta)$ be a $C^\infty$ map of the unit interval $[0,1]$ into
itself such that $\psi(\eta) =1$ for $0\leq\eta\leq \frac{1}{4}$ and
$\psi(\eta) = 0$ for $\frac{1}{2}\leq \eta\leq 1$. Let
$\Gamma(\xi,\eta,\mu) = (\xi, s(\eta), \mu)$, ie $\Gamma$ denotes the
change of coordinates $\eta \mapsto s(\eta)$ which leaves $\xi$ and
$\mu$ unchanged, where   $s(\eta)$ is a function like the one
described in \eqref{eq: s}.
Then $\phivec$ is
given by
\begin{equation}
\phivec_\perp  = (1 - \psi)
\lb\phivectilde_\perp\circ \Gamma\rb,\quad
\phi_\parallel = \lb 1 - |\phivec_\perp|^2\rb^\frac12.

\label{eq:phitilde2}
\end{equation}
From \eqref{n_jumpPi_small}
it follows that $|\phivectilde_\perp| \le
\epsilon_2 \le \half$ in $\Pi^b$, so that $\phi_\parallel$ is smooth.

The corresponding extenson
of $\tilde{\nvec}$ is given by
\begin{equation}
  \label{eq:ntilde2}
  \nvec(x,y,z) = \phivec\lb \frac{x}{\tau y},\frac{y}{W},
  \frac{z}{L}\rb.
\end{equation}
In analogy with \eqref{n_energyPi_newcoor},
the energy of $\nvec$ is given by
\begin{equation}
E[ \nvec ] = \int_{-\frac12}^{\frac12} d\mu \int_{-1}^1 d\xi
\int_{0}^{1}
\frac{d \eta}{\eta}
\lbr  \frac{L}{\tau} \lb \phivec_\xi \rb^2 +
L\tau \lb
\eta \phivec_\eta - \xi   \phivec_\xi \rb^2 +
\frac{W^2\tau}{L}
\eta^2
\lb
\phivec_\mu \rb^2 \rbr.
\label{n_energyLa_newcoor}
\end{equation}
Noting that $\phivec = \evec^b$ for $\eta\le \fourth $ (so that
integrand vanishes in this range),

one can obtain the estimate
\begin{equation}
  \label{eq:E_estimate_2}
E[ \nvec ] \le C^b_2 \int_{-\frac12}^{\frac12} d\mu \int_{-1}^1 d\xi
\int_{0}^{1}
d \eta
\lbr  L \lb \phivec_\xi \rb^2 +
L \lb \phivec_\eta\rb ^2 +
\frac{W^2}{L}
\lb
\phivec_\mu \rb^2 \rbr
\end{equation}
for some $C^b_2 > 0$ independent of $\nvectilde$, $\epsilon_1$,
$\epsilon_2$, $L$ and $W$ (in fact, it suffices to take $C^b_2 = 8\tau + 4\tau^{-1}$).

The derivatives of $\phivec$ can be expressed in terms of derivatives
of $\phivectilde$ as follows.  We have that $\phivec_{\perp\xi}
= (1-\psi)(\phivectilde_{\perp\xi}\circ\Gamma)$, while
$\phi_{\parallel\xi} =
(1-\psi)^2((\phitilde_\parallel\, \phitilde_{\parallel\xi})\circ\Gamma)/\phi_\parallel$,
where we have used $|\phivectilde_\perp|^2 + {\phitilde_\parallel}^2 =
1$.  Since $|1-\psi|^2 \le 1$ and $|(\phitilde_\parallel\circ
\Gamma)/\phi_\parallel| \le 1$, it follows that
\begin{equation}
  \label{eq:partial_xi bound}
  (\phivec_\xi)^2 = (\phivec_{\perp\xi})^2 + (\phi_{\parallel\xi})^2
\le (\phivectilde_\xi\circ\Gamma)^2.
\end{equation}
A similar argument shows that
\begin{equation}
  \label{eq:partial_mu bound}
  ( \phivec_\mu)^2 \le (\phivectilde_\mu\circ\Gamma)^2.
\end{equation}
Next, we have that
\begin{equation}
  \label{eq:partial_eta perp}
  \phivec_{\perp\eta} = -\psi'(\phivectilde_\perp\circ\Gamma) +
  (1-\psi)\,s'\,
(\phivectilde_{\perp\eta}\circ\Gamma),
\end{equation}
which implies that
\begin{equation}
  \label{eq:partial_eta perp est}
  (\phivec_{\perp\eta})^2 \le
  2{\psi'}^2|\phivectilde_\perp\circ\Gamma|^2 + 2{s'}^2(\phivectilde_{\perp\eta}\circ\Gamma)^2.
\end{equation}
Similarly,
\begin{equation}
  \label{eq:partial_eta para}
  \phi_{\parallel\eta} =
  (1-\psi)\psi'\lb|\phivectilde_\perp|^2
\circ\Gamma\rb/\phi_\parallel +
  (1-\psi)^2\,s'\,
((\phitilde_\parallel\,\phitilde_{\parallel\eta})\circ\Gamma)/\phi_\parallel.
\end{equation}
From \eqref{n_jumpPi_small}, $|\phivectilde_\perp|^2\le {\epsilon_2}^2
< \fourth$ and $|\phi_\parallel| >\half$, so that
\begin{equation}
  \label{eq:partial_eta para est}
  (\phi_{\parallel\eta})^2 \le
  2{\psi'}^2{\epsilon_2}^2 + 2{s'}^2(\phitilde_{\parallel\eta}\circ\Gamma)^2.
\end{equation}
Together, \eqref{eq:partial_eta perp est} and \eqref{eq:partial_eta
  para est} give
\begin{equation}
  \label{eq:partial_eta est}
  (\phivec_\eta )^2 \le 4 C_3 {\epsilon_2}^2 +
  2{s'}^2(\phivectilde_\eta\circ \Gamma)^2,
\end{equation}
where $C_3$ bounds ${\psi'}^2$.

We substitute \eqref{eq:partial_xi bound}, \eqref{eq:partial_mu bound}
and \eqref{eq:partial_eta est} into the estimate
\eqref{eq:E_estimate_2}.  Replacing $\int_0^1 d\eta$ by
$\int_{\frac12}^1\eta'(s)ds$, we get
\begin{equation}
  \label{eq:E est penultimate}
  E[\nvec] \le C^b_2 C_4
\lb
L {\|  \phivectilde_\xi \|}^2_{L^2(\Pi^b)} +
L \lb  4C_3 {\epsilon_2}^2 + 2{\|\phivectilde_\eta
  \|}^2_{L^2(\Pi^b)}\rb +
\frac{W^2}{L} {\|  \phivectilde_\mu \|}^2_{L^2(\Pi^b)}
\rb,
\end{equation}
where $C_4$ bounds both $s'$ and $\eta' = 1/s'$.  From
\eqref{tilde_phi_xi_norm},
\begin{equation}
  \label{eq:last line in lemma2}

      E[\nvec] \le 4C^b_1 C^b_2 C_4 \epsilon_1 + 4LC_2^bC_3C_4{\epsilon_2}^2,
\end{equation}
which implies the required result.
\end{proof}

\begin{thm}\label{thm: thm 2}
Suppose the tangent unit-vector field $\nvectilde$
is continuous on  $P$ with
$ {\| \nvectilde\|}^2_{1,2}(P) =   \int_P \lb\nabla
\nvectilde\rb^2 <\infty.$ Then for all  $\epsilon > 0$, there exists a
$C^\infty(P)$
 tangent unit-vector field $\nvec$  homotopic to
$\nvectilde$ with
$\|\nvectilde-\nvec\|_{1,2}(P) \leq \epsilon .$
\end{thm}

\begin{rem} In general, $\nvec(\rvec) -
\nvectilde(\rvec)$ is not uniformly small in $\rvec$; in small
neighbourhoods of the edges and vertices, this difference may be of
order one.  However, the contribution of these neighbourhoods to the
Sobolev norm, $\|\nvectilde-\nvec\|_{1,2}(P)$, is small.
\end{rem}
\begin{proof}
  {\em Step 0: truncation, reflection}.  Let $\Lambda^a(H)$ denote the
  vertex prisms defined in Lemma~\ref{lem: vertex extension}.  We
  choose $H$ sufficiently small so that these  do not intersect, and so
  that, for given $\epsilon_1 > 0$,
\begin{equation}\label{eq: nvectilde est on Lambda^a}
  \| \nvectilde\|^2_{1,2}\lb\cup_a\Lambda^a(H)\rb \leq  \epsilon_1.
\end{equation}

Let $\Lambda^b(W,\Ltilde^b)$ denote the edge prisms defined in
Lemma~\ref{lem: edge}.  We take these to have the same width, $W$, but
allow them to have different lengths $\Ltilde^b$.  We choose $W$
sufficiently small and the $\Ltilde^b$'s so that the following
conditions are satisfied: First, the $\Lambda^b(W,\Ltilde^b)$'s do not
interesect.  Second, the top and bottom faces of
$\Lambda^b(W,\Ltilde^b)$ (given by $z = \pm \half\Ltilde^b$ in the
local Euclidean coordinates of (\ref{eq: local edge coords})) are
contained in the respective vertex prisms
$\Lambda^a(\smallfrac{3}{12}H)$ and $\Lambda^{a'}(\smallfrac{3}{12}H)$
at the endpoints of $E^b$, but do not intersect the smaller vertex
prisms $\Lambda^a(\smallfrac{2}{12}H)$ and
$\Lambda^{a'}(\smallfrac{2}{12}H)$.
Third, for given $\epsilon_1 > 0$,
\begin{equation}
  \label{eq:condition 3}
  \| \nvectilde\|_{1,2}^2\lb\cup_b\Lambda^{b}(W,\Ltilde^b)\rb \leq
\half\epsilon_1.
\end{equation}
Fourth, for given $0 < \epsilon_2 < \half$,
\begin{equation}
  \label{eq:condition 4}
  \max_{\rvec\in \Lambda^b(W,\Ltilde^b)} \lv {\nvectilde}(\rvec)
  - \evec^b\rv \le \half \epsilon_2 < \fourth,
\end{equation}
where $\evec^b$ is
  the edge orientation of ${\nvectilde}$ on $E^b$.

Let
\begin{equation}
  \label{eq:Ptilde}
  P'= \overline{P
\setminus\lb \cup_a
\Lambda^{a}(\smallfrac{4}{12} H)\cup_b \Lambda^{b}(\half W, \Ltilde^b)\rb}
\end{equation}
be the closed polyhedron obtained by removing vertex and edge prisms
of indicated size from $P$.  Choose $\Delta_1 > 0$ sufficiently small
so that, for all $\rvec \in P'$, $B(\rvec,\Delta_1)$ -- the
$\Delta_1$-ball centered at $\rvec$ -- intersects at most one face
of $P$.  Let
\begin{equation}
  \label{eq:P' + delta_1}
  P' + \Delta_1 = \lbr \rvec\in \Rr^3 \big |\, |\rvec - \rvecp| \le \Delta_1
  \ \text{for some}\ \rvecp \in P' \rbr
\end{equation}
denote the closed $\Delta_1$-neighbourhood of
$P'$.  We define a unit-vector field $\nvectilde_+$ on
$P' + \Delta_1$ as follows.  If $\rvec \in P$,
$\nvectilde_+(\rvec)$ is just taken to be $\nvectilde(\rvec)$.  If
$\rvec \in (P' + \Delta_1)\setminus P$, then we must have that
$\rvec = \pvec + \alpha \Fvec^c$ for some (uniquely determined) $\pvec$
on a face $F^c$ of $P$
and for some $0 <\alpha \le \Delta_1$.  In this case, we take
$\nvectilde_+(\pvec + \alpha \Fvec^c) = {\cal R}\,\cdot\, \nvectilde_+(\pvec -
\alpha \Fvec^c)$, where ${\cal R}$ denotes reflection about the plane
normal to
$\Fvec^c$.
It is clear
that the Sobolev norm $\| \nvectilde_+
\|_{1,2} (P' + \Delta_1)$ of the extended vector field is finite.
Tangent boundary conditions imply that
$\nvectilde_+$ is continuous at points $\pvec\in P'$
that lie on a  face of $P$.
Moreover,
the average of $\nvectilde_+$ over
$B(\pvec,\delta)$, where $\delta < \Delta_1$,
is tangent
to the face.  This remains true
for weighted averages over $B(\pvec,\delta)$ provided the weights at $\pvec
+ \alpha \Fvec^c$ and $\pvec
- \alpha \Fvec^c$ are equal.\\

\noindent {\em Step 1: Bulk.}
Since $\nvectilde$ is continuous
on $P$, $\nvectilde_+$ is continuous on $P'+\Delta_1$, and
therefore uniformly continuous (since $P'+\Delta_1$ is compact).
Choose $\Delta_2 >
0$ sufficiently small so that
 \begin{equation}
 \rvec, \rvecp \in P'+ \Delta_1, \quad |\rvec - \rvecp| \le
 \Delta_2 \implies
 \lv \nvectilde_+(\rvec) -
 \nvectilde_+(\rvecp)\rv \leq \fourth \epsilon_2.
 \end{equation}
Let $K(r)\geq 0$ be a smooth function with support contained in
$(0,1)$, normalized so that
\begin{equation}
  \label{eq:K(r)}
  4\pi \int_0^\infty K (r) r^2 dr =1.
\end{equation}
Choose $\delta>0$ so that $\delta < \min(\Delta_1,\Delta_2).$
We construct a smooth unit-vector field $\uvec$  on $P'$ by averaging
$\nvectilde_+$ over balls of radius $\delta$, as follows:
\bge
\uvec(\rvec) =
\int_{\Rr^3}
\frac{1}{\delta^3} K\lb \frac{\lv \rvec - \rvecp \rv}{\delta}\rb
\nvectilde_{+}(\rvecp) \, dV'.\label{eq:uvec}
\ene
Uniform continuity ensures that $|\uvec(\rvec) - \nvectilde(\rvec)| \le
\fourth\epsilon_2< \smallfrac{1}{8}$ for  $\rvec \in P'$, and in particular that
$\uvec \ne 0$.  Therefore, the unit-vector field
\begin{equation}
  \label{eq:vvec}
  \vvec := \uvec/|\uvec|
\end{equation}
is smooth, with
\begin{equation}
  \label{eq:vvec - ntilde on P'}
 \max_{\rvec\in P'} | \vvec(\rvec) - \nvectilde(\rvec)| \le
\half\epsilon_2.
\end{equation}
$\vvec$ satisfies tangent boundary conditions at points $\pvec$ in $P'$
on faces $F^c$ of $P$ (since $K(\alpha \Fvec^c/\delta) =
K(-\alpha \Fvec^c/\delta)$).

It is straightforward to show (the arguments are similar to those of the
Meyers-Serrin theorem \cite{adams}, see also Appendix A, \cite{bl})
that
$\|\vvec - \nvectilde \|_{1,2} (P')$
can be made arbitrarily small
with $\delta$.  Indeed, from \eqref{eq:uvec}, for $\rvec \in P' $,
\begin{multline}
  \label{eq:pointwise grad difference}
  \nabla \uvec(\rvec) -
\nabla  \nvectilde(\rvec) =
\int_{\Rr^3}
\frac{1}{\delta^3}K\lb \frac{\lv \rvec - \rvecp \rv}{\delta}\rb
\lb
\nabla\nvectilde_{+}(\rvecp) -
\nabla\nvectilde_{+}(\rvec)\rb
\, d^3 {r'} = \\
= \int_{\Rr^3}
\ls
\frac{1}{\delta^{3/2}}
K^\half\lb \frac{\lv \rvec - \rvecp \rv}{\delta}\rb \rs
\ls
\frac{1}{\delta^{3/2}}
K^{\smallfrac{1}{2}}\lb \frac{\lv \rvec - \rvec' \rv}{\delta}\rb
\lb \nabla\nvectilde_{+}(\rvecp) -
\nabla\nvectilde_{+}(\rvec)\rb
\rs
\, d^3 {r'}.
\end{multline}
The Cauchy-Schwartz inequality gives that
\begin{equation}
  \label{eq:last in line}
|\nabla \uvec(\rvec) -
\nabla \nvectilde(\rvec)|^2 \le \int_{\Rr^3}\frac{1}{\delta^3}
K\lb \frac{\lv \rvec - \rvecp \rv}{\delta}\rb
\lv
\nabla \nvectilde_+(\rvec) -
\nabla \nvectilde_+(\rvecp)
\rv^2 \, d^3 r'.
\end{equation}
 Since  square-integrable vector fields can be
approximated arbitrarily closely (with respect to the $L^2$-norm) by
continuous vector fields, we may write $\nabla \nvectilde_+ ={\mathbf c} + {\mathbf h},
$ where ${\mathbf c}$ is continuous, and therefore uniformly continuous, on $P' +
\Delta_1$, and $\| {\mathbf h}\|_{L^2}(P'+\Delta_1)$
is arbitrarily small. Extend
${\mathbf h}$ by zero outside $P'+\Delta_1.$ Then
\begin{multline}
\|\nabla \uvec -
\nabla \nvectilde_+\|^2_{L^2}(P')\leq
\int_{P'}  d^3 r \int_{\Rr^3}  d^3r' \frac{1}{\delta^3}
K\lb \frac{\lv \rvecp - \rvec \rv}{\delta}\rb
\lv
\nabla \nvectilde_+(\rvec) -
\nabla \nvectilde_+(\rvecp)
\rv^2 \leq \notag \\
\leq  \int_{P'}  d^3 r \int_{\Rr^3} d^3 r'  \frac{1}{\delta^3}
K\lb \frac{\lv \rvec - \rvecp \rv}{\delta}\rb
3\lb  |{\mathbf h}(\rvec)|^2 + |{\mathbf h} (\rvecp)|^2 + \lv
 {\mathbf c}(\rvec) -
 {\mathbf c}(\rvecp)
\rv^2
\rb \leq
\notag \\
\leq
 3 \| {\mathbf h}\|^2_{L^2}(P'+\Delta_1) + 3 \| {\mathbf
  h}\|^2_{L^2}(P') +
3 \int_{P'} d^3 r\int_{\Rr^3}d^3 r' \frac{1}{\delta^3}
K\lb \frac{\lv \rvec - \rvecp \rv}{\delta}\rb
\lv
 {\mathbf c}(\rvec) -
 {\mathbf c}(\rvecp)
\rv^2.
\end{multline}

The first two terms can be made arbitrarily small, and since ${\mathbf c}$ is uniformly
continuous, the last term approaches zero with $\delta$.  That the
same is true for $\vvec = \uvec/|\uvec|$ is easily established; note
that
$\nabla \lv
\uvec \rv^2 = \nabla \lb(\uvec-
\nvectilde_+)\cdot(\uvec + \nvectilde_+)\rb$,
and  that $(\uvec + \nvectilde_+)$ is uniformly
bounded on $P'$ while $( \uvec-
\nvectilde_+)$ approaches zero uniformly with $\delta$.
Thus, we can choose
$\delta$ small enough so that
\begin{equation}
  \label{eq:vvec-ntilde}
  \|\vvec -
\nvectilde \|_{1,2}^2 (P')\le \half\epsilon_1.
\end{equation}
\\
\noindent {\em Step 2: Edges.} Let $\Lambda^{b}(W,L^b)$ be  edge
prisms shorter than those introduced in  Step 0, ie $L^b <
\Ltilde^b$, such that their top and bottom faces are contained in
$\cup_a \Lambda^a(\smallfrac{5}{12}H)\setminus\Lambda^a(\smallfrac{4}{12}H)$.
Let $\Pi^b = \overline{\Lambda^b(W,L^b)\setminus\Lambda^b(\frac12 W,L^b)}$ denote the
half-prism, as in Lemma~\ref{lem: edge}, and let
\begin{equation}
  \label{eq:Lambda and Pi}
  \Lambda_E = \cup_b \Lambda^b(W,L^b), \quad \Pi_E = \cup_b \Pi^b
\end{equation}
denote the union of the edge prisms and half-prisms respectively.
 $\vvec$ defines a  smooth tangent unit-vector field on
$\Pi_E$.
From \eqref{eq:condition 3} and \eqref{eq:vvec-ntilde}, its energy is
bounded by
\begin{equation}
\|\vvec\|_{1,2}^2(\Pi_E)
\leq
2\|\vvec - \nvectilde  \|^2_{1,2}(\Pi_E) +
2\| \nvectilde  \|^2_{1,2} (\Pi_E) \leq
\epsilon_1+  \epsilon_1 \leq 2\epsilon_1.
\end{equation}
From \eqref{eq:condition 4} and \eqref{eq:vvec - ntilde on P'},
\begin{equation}
  \label{eq:second condition of lemma 2}
\max_{\rvec \in \Pi^b}
\left|
\vvec(\rvec) -
\evec^b \right| \le
\max_{\rvec \in \Pi^b}
\lb
\left|
\vvec(\rvec) -
\nvectilde(\rvec) \right| +
\left|
\nvectilde(\rvec) -
\evec^b \right|

\rb
\leq \half \epsilon_2+ \half \epsilon_2 = \epsilon_2.
\end{equation}
Thus, the conditions of Lemma~\ref{lem: edge} are  satisfied for each
$\Lambda^b(W,L^b)$, and
we can construct a smooth tangent unit-vector field $\wvec$ on
$\Lambda_E$ which coincides with $\vvec$ on
$\Lambda_E\setminus(\cup_b \Lambda^b(\smallfrac{3}{4} W, L^b))$
such that
\begin{equation}
\label{eq: wvec small}
\|\wvec \|_{1,2}^2 \lb \Lambda_E  \rb \leq
{\textstyle \sum_b} C^b ( 2 {\epsilon_1} + L^b{\epsilon_2}^2),
\end{equation}
where $C^b > 0$ is independent of $\epsilon_1$, $\epsilon_2$,
$\nvectilde$, $W$ and $L^b$.
We may extend $\wvec$ (smoothly) to
$  P'' = \overline{
P\setminus\lb \cup_a \Lambda^a(\smallfrac{5}{12} H)\rb}$
by taking $\wvec$ to coincide with
$\vvec$ on  $P''\setminus\Lambda_E$.
From \eqref{eq:condition 3},
\eqref{eq:vvec-ntilde}
and \eqref{eq: wvec small},
\begin{align} \label{eq: wvec - nvectilde estimate}
\|\wvec -\nvectilde \|_{1,2}^2 \lb P'' \rb &=
 \|\wvec -\nvectilde \|_{1,2}^2 \lb P''\setminus \Lambda_E \rb
+  \|\wvec -\nvectilde \|_{1,2}^2 \lb \Lambda_E \rb \nonumber
\\
&\leq \|\vvec -\nvectilde \|_{1,2}^2 \lb P''\setminus \Lambda_E \rb
+ 2\lb \|\wvec\|_{1,2}^2 + \|\nvectilde\|_{1,2}^2\rb \lb \Lambda_E \rb \notag
\\
&\leq \lb \smallfrac{3}{2} + 4{\textstyle \sum_b} C^b\rb \epsilon_1 + 2 {\textstyle \sum_b}
C^b L^b{\epsilon_2}^2.

\end{align}
In the second inequality we have used the fact that $P''\setminus
\Lambda_E \subset P'$.

{\em Step 3. Vertices.}
Let $\Pi^a = \overline{\Lambda^a(H)\setminus \Lambda^a(\frac12 H)}$ denote the half-prism,
as in Lemma~\ref{lem: vertex extension}, and let
\begin{equation}
  \label{eq:Lambda and Pi vertices}
  \Lambda_V = \cup_a \Lambda^a(H), \quad \Pi_V = \cup_a \Pi^a
\end{equation}
denote the union of the vertex prisms and half-prisms respectively.
From \eqref{eq: nvectilde est on Lambda^a} and
\eqref{eq: wvec - nvectilde estimate},
\begin{multline}
\| \wvec \|^2_{1,2} \lb \Pi_V \rb
\leq 2 \|\wvec -\nvectilde \|^2_{1,2} \lb \Pi_V \rb +
2 \|\nvectilde \|^2_{1,2} \lb \Pi_V \rb  \leq \\ \leq
\lb \lb 5  + 8{\textstyle \sum_b} C^b \rb \epsilon_1 +
4{\textstyle \sum_b}
C^b L^b{\epsilon_2}^2\rb
:= \epsilon_3.
\end{multline}
Thus, the conditions of Lemma~\ref{lem: vertex extension} are satisfied for
each $\Lambda^a(H)$, and
we can construct a smooth tangent unit-vector
field $\nvec$ on $\Lambda_V$ coinciding with $\wvec$ on
$\Lambda_V\setminus(\cup_a\Lambda^a(\smallfrac{3}{4} H)$ such that
\begin{equation} \label{eq: nvec est }
\|\nvec \|^2_{1,2} \lb\Lambda_V \rb \leq {\textstyle \sum_a} C^a
{\epsilon_3}
\end{equation}
where $C^a > 0$ is independent of $H$, $\epsilon_3$ and $\nvectilde$.

We extend $\nvec$ (smoothly) to all of $P$ by taking $\nvec$ to
coincide with $\wvec$ on $P\setminus\Lambda_V$.  From \eqref{eq: nvectilde est on Lambda^a},
\eqref{eq: wvec -
  nvectilde estimate} and \eqref{eq: nvec est },
\begin{align}

\|\nvec -\nvectilde \|_{1,2}^2
\lb P \rb
& =   \| \nvec -\nvectilde \|_{1,2}^2
\lb P\setminus \Lambda_V \rb
+  \|\nvec -\nvectilde \|_{1,2}^2
\lb \Lambda_V \rb \notag
\\
& \leq   \|\wvec -\nvectilde \|_{1,2}^2
\lb P\setminus \Lambda_V \rb
+ 2
\lb \|\nvec\|_{1,2}^2 + \|\nvectilde\|_{1,2}^2\rb
\lb \Lambda_V \rb \notag
\\
&\leq
\lb \half + 2{\textstyle \sum_a} C^a \rb
\epsilon_3 +  \epsilon_1.
\end{align}
By choosing $\epsilon_1$ and $\epsilon_2$ appropriately,
$\|\nvec -\nvectilde \|_{1,2}^2$ can be made arbitrarily small.
From \eqref{eq:vvec - ntilde on P'}, $\lv \nvec -\nvectilde \rv \leq \half \epsilon_2$
on $P'$.  This ensures that $\nvec$ and $\nvectilde$ have the
same kink numbers and wrapping numbers.  The construction also ensures
that their edge orientations are the same.
\end{proof}

\section{Discussion}\label{sec:discussion}

We have derived a lower bound, depending on homotopy type, for the
energies of harmonic maps of convex polyhedra in $\Rr^3$ to the unit
sphere $S^2$ which satisfy tangent boundary conditions and are
continuous everywhere except at vertices.
It is natural to ask whether this lower bound is sharp.  For the case
of a rectangular prism, eg a cube, numerical calculations and
analytical arguments have indicated that, for a large set of homotopy
types, the lower bound (\ref{low_bound}) is not sharp, but differs
from the actual lower bound by a fixed quantity independent of
homotopy type \cite{mrz}.  A related question is whether the infimum
(\ref{energyMinimimal}) is achieved by some $\nvec \in \czero(P)\cap
W^{1,2}(P)$.  Numerics and analytic arguments for a rectangular prism
have suggested that for a small number of ``unwrapped'' states, the
infimum is achieved, but for a large set of homotopy types, a
sequence of configurations with energies approaching $M(h)$ develop
discontinuities along the edges.  We plan to present these results in
a subsequent paper.

While our present motivation comes from liquid crystal physics,
related considerations for maps of two- (three-) dimensional polyhedra
to spheres may be relevant for string (M-) theories with tangent
boundary conditions (`$T$ branes').

\section*{Acknowledgment}\label{sec:acknowledgment}
We are grateful to Adrian Geisow and Chris Newton, Hewlett-Packard
Laboratories, Bristol for stimulating out interest in this area.
AM was supported by an
Industrial CASE studentship funded by EPSRC and by Hewlett-Packard
Laboratories, Bristol.


TITLE: Why is $GL_2(\mathbb C)$ connected?
QUESTION [4 upvotes]: I have proved that the set of all $2\times 2$ non-singular matrices over $\mathbb R$ is disconnected as $\{A:\det A>0\}$ and $\{A:\det A<0\}$ forms a disconnection of $GL_2(\mathbb R)$.
What I can't understand is that why is $GL_2(\mathbb C)$ connected?
How should I approach?

REPLY [2 votes]: This might be a bit of an overkill here, but I thought I would also mention an elegant proof coming the theory of Lie groups, using the existence of a logarithm for any invertible matrix. Let $a, b \in \operatorname{GL}_n(\mathbb{C})$, and pick $A, B \in \mathcal{M}_n(\mathbb{C})$ two logarithms of $a$ and $b$ respectively (which means here : matrices such that $e^A = a$ and $e^B = b$).
Then we can define a continuous path $\Gamma : t \in [0,1] \mapsto \exp(tB + (1-t)A)$ which is always in $\operatorname{GL}_n(\mathbb{C})$ (since the exponential of a matrix is always invertible) and goes from $a$ to $b$.


TITLE: Every operator $T : X \rightarrow L^1$ is weakly compact.
QUESTION [1 upvotes]: I am trying the following problem.
Let $X$ be a Banach space that does not contain a copy of $l^1$. Show that every operator $T : X \rightarrow L^1$ is weakly compact.
I am not sure how to start. Should we use the equi-integrability criterion? Any help is appreciated!

REPLY [1 votes]: Suppose $T:X\rightarrow L_1$ is not weakly compact. Then, using the Eberlein–Šmulian theorem, there is a bounded sequence $(x_n)$ in $X$ such that $F=\{ Tx_n : n=1,2,\ldots\}$ is not relatively weakly compact in $L_1$. This implies that $F$ contains a basic sequence $(T x_{n_k})_k$ that is equivalent to the standard unit vector basis of $\ell_1$ (by a well-known characterization of weakly compact sets in $L_1$. See e.g., Theorem 5.2.9 of Albiac and Kalton's Topics in Banach Space Theory).
Let $Z$ be the closed linear span of $\{ Tx_{n_k}: k=1,2,\ldots\}$. $Z$, of course, is $\ell_1$ is disguise with "standard basis" $(Tx_{n_k})$.
Let $Y$ be the closed linear span of the $x_{n_k}$ in $X$. Let's look at  $T|_Y$.
Note $T(Y)\subseteq Z$. We now show $T|_Y$ is in fact  onto $Z$. Let $z\in Z$. Then $z=\sum\limits_{k=1}^\infty \alpha_k Tx_{n_k}$ for some sequence $(\alpha_k)\in\ell_1$. Since $(\alpha_k)\in\ell_1$ and $(x_{n_k})$ is bounded,
$$y=\sum_{k=1}^\infty \alpha_k x_{n_k}=\lim_{m\rightarrow\infty}\sum_{k=1}^m\alpha_k x_{n_k}$$  is well-defined. Further $y\in Y$ and
$$Ty=\lim_{m\rightarrow\infty}\sum_{k=1}^m\alpha_k Tx_{n_k}=z.$$
So $T|_Y$ is onto $Z$.
Thus $T|_Y$ is a bounded operator from the Banach space $Y$ onto a space isomorphic to $\ell_1$. This implies (see exercise 2.8 in  Albiac-Kalton, e.g.) that $Y$, and thus $X$, contains a copy of $\ell_1$.


TITLE: Showing that $f(x) = \dfrac{x}{(2\ln x)^2}$ is an increasing function for $x \ge 8$
QUESTION [3 upvotes]: I apologize for the repetition.  I asked a similar question here before.
I was trying to generalize the result.  Does the following reasoning also work to show that $f(x) = \dfrac{x}{(2\ln x)^2}$ is an increasing function for $x \ge 8$
Please let me know if any of these steps are wrong:
(1)  Using the quotient rule with $g(x) = x$ and $h(x) = (2\ln x)^2$:
$$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$$
(2) Using the exponent rule for derivatives with $s(x) = 2\ln x$:
$$h'(x) = (s(x)^2)' = s(x)^2\left(s'(x)\dfrac{2}{s(x)}\right) = 2s(x)s'(x)$$
(3) $s'(x) = \dfrac{2}{x}$ so that:
$$h'(x) = \dfrac{8\ln(x)}{x}$$
(4) With $g'(x) = 1$, it follows that:
$$f'(x) = \dfrac{(2\ln x)^2 - \frac{8x\ln(x)}{x}}{(2\ln x)^4} = \dfrac{(2\ln x) - 4}{(2\ln x)^3}$$
(5)  It is increasing at $x\ge 8$ since:
$$\dfrac{(2\ln 8) - 4}{(2\ln(8))^3} > 0.0022 > 0$$
Are these steps correct?

Edit:  I changed step(5) to $x\ge 8$ since that is my goal.
It looks like my result may be correct for $x=8$ but insufficient for $x \ge 8$.

Edit 2:  Made a fix based on John Omielan's comment.

REPLY [1 votes]: You have almost reached there. You will just have to modify your step 5.
$$f'(x)=\frac{2(ln\ x-2)}{8(ln\ x)^3}$$
$$f'(x)=\frac{ln\ x-2}{4(ln\ x)^3}$$
We know that $ln\ x>0\  \forall x>1$ and hence for all $x\ge8$. Therefore its cube, $(ln\ x)^3$ is also positive $\forall x\ge8$. That makes the denominator of $f'(x)$ positive $\forall x\ge8$.
Also, $ln\ x$ is a strictly increasing function as its first derivative, $\frac{1}{x}$ is positive in all its domain, i.e., $\forall x>0$.
We also know that $ln\ 8=2.079... >2$, and by the definition of an increasing function we can conclude that $ln\ x\ge ln\ 8>2, \forall x\ge 8$.
$$\implies ln\ x-2>0, \forall x\ge8$$
Hence, we also established that the numerator of $f'(x)$ is positive $\forall x\ge8$.
A rational function is positive if and only if its numerator and denominator functions have the same sign.
Since both its numerator and its denominator are always positive, and hence have same sign, $\forall x\ge8$ we can conclude that $f'(x)$ is positive in the same interval.
Hence proved that $f(x)$ is increasing $\forall x\ge8$


TITLE: Example 3 Sobolev spaces Evans PDE
QUESTION [3 upvotes]: In the following example p.260 Evans.
Can someone explain me the part that says if $\alpha + 1 < n, |Du(x)| \in L^1(U)$
And how do i get the middle inequality in the next line.
Thanks in advance

REPLY [5 votes]: It is an integrability condition, in particular it is the $n$-dimensional analogue to the statement  $\int_0^1\frac{1}{x^p}\,dx < \infty$ if $p < 1$. This condition is readily obtained by the Coarea formula: $$\int_{B(0,1)}\frac{1}{|x|^p}\,dx = \int_0^1\int_{\partial B(0,r)}\frac{1}{r^p}\,dS\,dr = \int_0^1\frac{1}{r^p}C(n)r^{n-1}\,dr = C(n)\int_0^1r^{-(p +1-n)}\,dr.$$ This proves that $|x|^{-p}$ is in $L^1(B(0,1))$ provided $p < n$. Hence, since $|Du(x)| = \alpha|x|^{-(\alpha + 1)}$, we obtain the condition $\alpha + 1 < n$.
Notice that, by definition, $u$ restricted to $\partial B(0,\epsilon)$ is constant and equal to $\epsilon^{-\alpha}$. Moreover $|\nu^i| \le |\nu| = 1$ since $\nu$ is the unit inward pointing normal to $\partial B(0,\epsilon)$. Hence one has
\begin{align}
\Big|\int_{\partial B(0,\epsilon)}u\phi\nu^i\,dS\Big| \le &\ \int_{\partial B(0,\epsilon)}|u\phi\nu^i|\,dS \le \int_{\partial B(0,\epsilon)}\|\phi\|_{\infty}|\nu||u|\,dS \\
= &\ \|\phi\|_{\infty}\epsilon^{-\alpha}\int_{\partial B(0,\epsilon)}\,dS.
\end{align}


text:
\section{Topology is relevant}
  \label{sec:topology}

The original CSP dichotomy conjecture for reducts of finitely bounded homogeneous structures due to Bodirsky and Pinsker (see~\cite{BPP-projective-homomorphisms}) \changed{envisions} the following borderline between tractability and hardness.

\begin{conjecture}\label{conj:tract}
Let $\aA$ be a~reduct of a~finitely bounded homogeneous structure. Exactly one of the following holds:
\begin{enumerate}
  \item some stabilizer of the polymorphism clone of its model-complete core possesses a~continuous clone homomorphism to $\Proj$, and $\CSP(\aA)$ is \NP-complete,
  \item no stabilizer of the polymorphism clone of its model-complete core possesses a~continuous clone homomorphism to $\Proj$, and $\CSP(\aA)$ is in \Ptime.
\end{enumerate}
\end{conjecture}

Barto and Pinsker showed in~\cite{BartoPinskerDichotomy, Topo} that topology is irrelevant in this conjectured borderline, since the word `continuous' can simply be dropped without changing the \changed{force of} the conjecture. More precisely, when $\aA$ is any $\omega$-categorical structure with model-complete core $\bB$, then some stabilizer of $\Pol(\bB)$ possesses a~clone homomorphism to $\Proj$ if and only if some stabilizer of $\Pol(\bB)$ possesses a~continuous such homomorphism, and this is witnessed by the non-satisfaction of the pseudo-Siggers identity in $\Pol(\bB)$.

Following the discovery of the importance of minion homomorphisms for the complexity of CSPs in~\cite{BOP18}, it was then shown that whenever $\aA$ is any $\omega$-categorical structure with less than doubly exponential orbit growth (a condition satisfied in particular by all structures in the range of the conjecture), then the above hardness criterion is equivalent to the existence of a~uniformly continuous minor preserving map from $\Pol(\aA)$ to $\Proj$~\cite{BKOPP17, BKOPP17a}.

Naturally, the question of whether topology is irrelevant also for minion homomorphisms was raised in this context~\cite{BartoPinskerDichotomy, BOP18, Topo}, in particular for $\omega$-categorical structures with less than doubly exponential orbit growth.

\begin{question} \label{quest:topo}
Let $\aA$ be an $\omega$-categorical structure.
If there exists a~minion homomorphism $\Pol(\aA)\to\Proj$, does there exist a~uniformly continuous one?
\end{question}

While a~positive answer was obtained in some special cases~\cite{BKOPP17, BKOPP17a}, we are going to provide a~negative answer to the question in general.
The remainder of this section will be devoted to the construction of the structure $\mathbb S$ and the verification of the properties claimed in Theorem~\ref{thm:main-3}.

\subsection{Encoding graphs in higher arities}

Our first step will be a~standard construction which allows us to encode graphs as structures on $n$-tuples, for arbitrary $n\geq 1$.

\begin{lemma}\label{lem:S}
  Let $\gG$ be a~finite connected loopless graph and $n \geq 1$. Then there exists a~structure $\sS(\gG,n)$ with a~single relation $R$ of arity $2n$ such that
  \begin{enumerate}
    \item The expansion $(\sS(\gG,n),\neq)$ of $\sS(\gG,n)$ by the inequality relation is an $\omega$-categorical model-complete core without algebraicity;
    \item $\sS(\gG,n)$ pp-constructs the Cherlin-Shelah-Shi structure $\CSS(\gG)$;
    \item The relation $R$ of\/ $\sS(\gG,n)$ only contains tuples with pairwise distinct entries;
    \item $\Aut(\sS(\gG,n))$ has at most $3^{k^{n|\gG|}}$ orbits of $k$-tuples, for every $k \geq 2$.
  \end{enumerate}
\end{lemma}

\begin{proof}
  The structure $\sS(\gG,n)$ is itself obtained via a~$\CSS$ structure for a finite family $\mathcal G$ of structures. Let $\gG'$ be the structure obtained from $\gG$ by replacing each vertex $x$ of $\gG$ by an $n$-tuple $\overline x$ of new distinct elements, and requiring, for vertices $x,y$ of $\gG$, the $2n$-ary relation $R(\overline x, \overline y)$ to hold if $(x,y)$ is an edge in $\gG$.  Note that $\gG'$ is connected because $\gG$ is connected.

Let $\mathcal G$ contain $\gG'$ as well as every connected structure on $<2n$ elements and containing a single $R$-tuple (such a structure is called \emph{loop-like} in the following).
Let $\mathbb{F}'$ be the $\CSS$ structure for $\mathcal G$.
Then $\mathbb{F}'$ is $\omega$-categorical and has no algebraicity. We set $(\sS(\gG,n),\neq)$ to be the model-complete core of the structure $(\mathbb{F}',\neq)$; it is also $\omega$-categorical~\cite{cores-journal}, and it follows from its construction that it has no algebraicity either (cf.\ the proof of Theorem~27 in~\cite{MMSNP}). Hence, item~(1) is satisfied.

We now show item~(2).  Let $\mathbb{T}$ be the graph whose vertices are the $n$-tuples of $\sS(\gG,n)$, and where two tuples $(x_1,\dots,x_n)$ and $(y_1,\dots,y_n)$ are related if and only if $R(x_1,\dots,x_n,y_1,\dots,y_n)$ holds in $\sS(\gG,n)$. Clearly, $\mathbb{T}$ is a~pp-power of $\sS(\gG,n)$. We claim that
  $\mathbb{T}$ and $\CSS(\gG)$ are homomorphically equivalent; this implies that $\CSS(\gG)$ is pp-constructible from $\sS(\gG,n)$, as required.

  To prove the claim, by a~standard compactness argument it is sufficient to show that a~finite graph homomorphically maps into $\mathbb{T}$ if and only if it homomorphically maps into $\CSS(\gG)$; in other words, a~finite graph homomorphically maps into $\mathbb{T}$ if and only if it does not contain a~homomorphic image of $\gG$. Suppose first that there existed a~homomorphism from $\gG$ into $\rel T$. Then $\gG'$ constructed as above would have a~homomorphism into $\sS(\gG,n)$, and hence also into $\rel F'$, a~contradiction.  Conversely, if $\hH$ is a~graph which does not contain a~homomorphic image of $\gG$, then $\gG'$ does not homomorphically map into $\hH'$, and therefore $\hH'$ embeds into $\rel F'$, and hence homomorphically maps into $\sS(\gG,n)$. But this implies that $\hH$ homomorphically maps to $\rel T$.

  Item $(3)$ of the lemma holds since we have included loop-like obstructions in the definition of $\rel F'$, and since $\rel F'$ and $\sS(\gG,n)$ are homomorphically equivalent.

  To see item~(4), note that the orbit-growth of a~homogeneous structure with relations of arity at most $r$ is bounded by $3^{k^r}$ for large enough $k$.
  By Theorem~\ref{thm:hubicka-nesetril}, $\CSS(\gG')$ has a~homogeneous expansion by relations with arity at most $|\gG'|=n|\gG|$.
  Thus, $\Aut(\mathbb{F}')$ has for large $k$ at most $3^{k^{n|\gG|}}$ orbits of $k$-tuples.
  Whence, the same holds for the model-complete core $\sS(\gG,n)$, which has at most the number of orbits of the original structure {(see~\cite{Bodirsky-HDR}, Proposition 3.6.24)}.

  \end{proof}

\subsection{Superposition of the encodings}
It is well-known that if two $\omega$-categorical structures $\mathbb A$ and $\mathbb B$ in disjoint signatures $\sigma$ and $\tau$ have no algebraicity,
then there exists a~generic superposition $\aA\odot \bB$ of the two in the signature $\sigma\cup\tau$ and which is unique up to isomorphism.

This generic superposition is again $\omega$-categorical and without algebraicity.
It is obtained as follows:
\begin{enumerate}
	\item Expand $\aA$ by all relations that have a first-order definition in $\aA$, and similarly for $\bB$. Call $\aA'$ and $\bB'$ the resulting structures and let $\sigma'$ and $\tau'$ be their signatures (that
	we take to be disjoint without loss of generality).
	\item Since $\aA$ and $\bB$ are without algebraicity, so are $\aA'$ and $\bB'$ (expanding by first-order definable relations does not change the automorphism groups of the structures).
	Thus, the class of finite substructures of $\aA'$ and $\bB'$ have the strong amalgamation property (see Proposition 2.15 in~\cite{Oligo}).
	\item The class of finite $(\sigma'\cup\tau')$-structures whose $\sigma'$- and $\tau'$-reducts embed into $\aA'$ and $\bB'$, respectively, has the strong amalgamation property,
	and we call $\aA'\odot \bB'$ its Fra\"iss\'e limit.
	The $(\sigma\cup\tau)$-reduct of $\aA'\odot\bB'$ is then our structure $\aA\odot\bB$.
\end{enumerate}
As an example, take $\aA$ to be $(\mathbb Q,<)$ and $\bB$ to be the random graph (i.e., the graph $\CSS(\mathbb L)$ where $\mathbb L$ is a graph on a single vertex with a loop).
Then $\aA\odot \bB$ is the \emph{random ordered graph}, i.e., the Fra\"iss\'e limit of the class of finite simple graphs with a~total ordering on the vertices.

The same construction works for generic superpositions of infinitely many $\omega$-categorical structures without algebraicity.
The generic superposition will have an infinite signature, but will be $\omega$-categorical if the Fra\"{i}ss\'{e} class which yields the superposition has finitely many inequivalent atomic formulas of each arity.

In our construction of the structure of Theorem~\ref{thm:main-3}, we would like to superpose the graphs {from Theorem~\ref{thm:chain}}; this superposition would however not be $\omega$-categorical as there would be infinitely many orbits of pairs of vertices. This is why we superpose encodings of these graphs on tuples of increasing arity instead.

\begin{construction}\label{constr:s}
Let $\hH_1,\hH_2,\dots$ be an enumeration of

{all non-3-colorable finite graphs (in fact, the graphs from the chain in Theorem~\ref{thm:chain} are sufficient).}
Let $g\colon\mathbb N\to\mathbb N$ be a \changed{strictly} increasing function. {We define $\sSa$ as the generic superposition of all of the structures $\sS(\hH_{n},g(n))$, for $n \geq 1$:}
\[
  \sSa:={\bigodot}_{n\geq 1} \sS(\hH_{n},g(n))\;.
\]
{Note that $\sSa$ depends on the enumeration of the graphs above as well as on the function $g$. The former dependence will be irrelevant for our purposes, but the latter will play a role, and we make the convention that in all statements we state all properties required of $g$ in order for them to hold.}
\end{construction}

We note that by Theorem~\ref{thm:hubicka-nesetril}, each $\sS(\hH_n,g(n))$ has an expansion by finitely many relations which is homogeneous.
The structure obtained by expanding $\sSa$ by this infinite set of relations is itself homogeneous.
In the proof below, we call this expansion `the' homogenization of $\sSa$, even though it is not unique.

\begin{lemma}\label{lem:core}
The structure $(\sSa,\neq)$ is an $\omega$-categorical model-complete core without algebraicity.
\end{lemma}
\begin{proof}
The generic superposition of structures without algebraicity {never has} algebraicity,
and expanding a~structure by $\neq$ does not introduce algebraicity since $\sSa$ and $(\sSa,\neq)$ have the same orbits.

We prove that $\sSa$ is $\omega$-categorical (which implies that $(\sSa,\neq)$ is $\omega$-categorical, by the sentence above).
First, we prove that every atomic formula $\phi(x_1,\dots,x_r)$ over $\sS(\hH_n,g(n))$
is either equivalent to ``false'' or has at least $g(n)$ different variables.
Suppose that $\phi$ is the relation symbol $R_n$ (and thus $r=2g(n)$). By construction of $\sS(\hH_n,g(n))$, since all loop-like structures have been forbidden,
we have that either all the variables are distinct, or $\phi(x_1,\dots,x_r)$ is not satisfiable in $\sS(\hH_n,g(n))$ and is equivalent to false.
Suppose now that $\phi$ is a~relation symbol added for the homogenization of $\sS(\hH_n,g(n))$.
Let $\hH_n'$ be the structure obtained from $\hH_n$ as in the proof of Lemma~\ref{lem:S}.
Note that the cuts of $\hH'_n$ have size at least $g(n)$, so we know from Theorem~\ref{thm:hubicka-nesetril} that $r\geq g(n)$.
Moreover, Theorem~\ref{thm:hubicka-nesetril} gives that $\phi$ is equivalent to a~pp-formula over $\sS(\hH_n,g(n))$. Then at least $g(n)$ of the variables of $\phi$ are different,
for otherwise a~clause in $\phi$ would be of the form $R_n(y_1,\dots,y_{2g(n)})$ with fewer than $2g(n)$ distinct variables, and $\phi$ would be equivalent to ``false''.
In conclusion, we obtain that all non-trivial atomic formulas over the homogenization of $\sS(\hH_n,g(n))$ have arity at least $g(n)$, and there are only finitely many of them since this homogenization
has a~finite signature.
Thus, since $g$ is an increasing function, the homogenization of $\sSa$ has only finitely many atomic formulas of each arity.
It follows that the homogenization of $\sSa$ has finitely many orbits of each arity, so that this homogenization is $\omega$-categorical, and thus $\sSa$ itself is $\omega$-categorical.

To see that $(\sSa,\neq)$ is a~model-complete core, let $e$ be an endomorphism of $(\sSa,\neq)$, and let $F$ be a~finite subset of its domain. Then $e$ is also, in particular, an endomorphism of $(\sS(\hH_n,g(n)),\neq)$ for all $n\geq 1$, and since the latter structures are model-complete cores, the restriction \changed{$e|_F$} of $e$ to $F$ has an expansion to an automorphism of $\sS(\hH_n,g(n))$ for each $n\geq 1$. It then follows that \changed{$e|_F$} is a~partial isomorphism of the Fra\"{i}ss\'{e} structure $\sS'$ of which $\sSa$ is the reduct. By homogeneity, \changed{$e|_F$} extends to an automorphism of $\sSa'$, which is also an automorphism of $\sSa$ and of $(\sSa,\neq)$.
\end{proof}

We show in the next lemma that the orbit growth of $\sSa$ can be controlled by picking a~suitable \changed{function} $g$ in the construction.

\begin{lemma}\label{lem:orbitgrowth}
For every increasing $f\colon\mathbb N\to\mathbb N$ that dominates every polynomial,
there exists a function $g \colon\mathbb N\to\mathbb N$ such that the number of orbits of $k$-tuples of\/ $\sSa$ is not asymptotically larger than \[3^{f(k)}.\]
In particular, there exists a function $g \colon {\mathbb N} \to {\mathbb N}$ such that $\sSa$ has less than doubly exponential orbit growth.
\end{lemma}
\begin{proof}
We construct $g$ by induction, first setting $g(1)=1$.
Suppose now that $g(1),\dots,g(n)$ are defined.
Since $f$ dominates every polynomial, there exists a~$k_n>g(n)$ such that $\sum_{i=1}^{n} k_n^{g(i)|\hH_i|} < f(k_n)$ for all $k\geq k_n$.
Let $g(n+1):=k_n+1$.

\changed{Let $\sSa$ be the structure associated with the function $g$ thus defined.}
Let $n\geq 1$. Orbits of $k_n$-tuples in $\sSa$ are uniquely determined by orbits of $k_n$-tuples in $\sS(\hH_m,g(m))$ for $m\leq n$;
this follows from the fact that $k_n<g(m)$ for $m>n$ and that the orbits of $k$-tuples of $\sS(\hH_m,g(m))$ are that of the empty structure if $k<g(m)$.
By Lemma~\ref{lem:S}, the number of orbits of $k_n$-tuples in $\sSa$ is at most $$3^{k_n^{g(1)|\hH_1|}}\cdots 3^{k_n^{g(n)|\hH_{n}|}} = 3^{\sum k_n^{g(i)|\hH_i|}} < 3^{f(k_n)}.$$
Therefore, the number of orbits of $\sSa$ is bounded above by $3^{f(k)}$ infinitely often.

To prove the final remark,

{let $f$ be the function given by $f(k) := 3^{\sqrt k}$,
which dominates every polynomial. Let $g \colon {\mathbb N} \to {\mathbb N}$ be the function
obtained from the statement applied to this $f$. Then the orbit growth of $\sSa$ is asymptotically at most $3^{3^{\sqrt{k}}}$, and hence less than doubly exponential. }
\end{proof}

\subsection{Identities in $\sSa$}

\changed{In the present section, $g\colon\mathbb N\to\mathbb N$ is a strictly increasing function, and $\sSa$ denotes the corresponding structure (Construction 5.4).}

\begin{lemma}\label{lem:noh1}
  $\Pol(\sSa)$ does not satisfy any non-trivial height~1 condition.
\end{lemma}
\begin{proof}
  For each non-trivial height~1 condition $\Sigma$ there exists an $n\geq 1$ such that $\Sigma_{\hH_n}$ is weaker than $\Sigma$.
  Since $\CSS(\hH_n)$ does not satisfy $\Sigma_{\hH_n}$, and $\CSS(\hH_n)$ is pp-constructible from $\sS(\hH_n,g(n))$, Theorem~\ref{thm:bop18} implies that the latter does not satisfy $\Sigma_{\hH_n}$ either, and in particular does not satisfy $\Sigma$.
  Therefore, $\Pol(\sSa)$ does not satisfy~$\Sigma$.
\end{proof}
Since $\Pol(\sSa,\neq)\subseteq\Pol(\sSa)$, we obtain in particular that $\Pol(\sSa,\neq)$ does not satisfy any non-trivial height~1 condition either.

We are not going to show directly that $(\sSa,\neq)$ satisfies non-trivial height~1 conditions locally, but will \changed{find another (not height~1) condition} it satisfies, and then use its slow orbit growth {(for suitable $g$)} to deduce the satisfaction of local height 1 conditions.

A polymorphism $f$ of a~structure is called a~\emph{pseudo-Siggers} operation if there are endomorphisms $e_1,e_2$ of the structure  such that for all $x,y,z$ of the domain
\[
  e_1\circ f(x,y,x,z,y,z) = e_2\circ f(y,x,z,x,z,y).
\]

\begin{lemma}\label{lem:psiggers}
  The structure $(\sSa,\neq)$ has a~pseudo-Siggers polymorphism.
\end{lemma}
\begin{proof}
For each $n\geq 1$, let $\sS_n$ be the generic superposition
$$
\sS(\hH_1,g(1))\odot \cdots\odot \sS(\hH_n,g(n))\; .
$$
Then $\Pol(\sS_n)$ satisfies a~quasi near unanimity identity of some sufficiently large arity. To see this, note that there exists $\ell\geq 1$ such that $\CSS(\hH_1),\ldots, \CSS(\hH_n)$ all have a~quasi near unanimity polymorphism of arity $\ell$, by Lemma~\ref{lem:fg}.
Similarly, such an $\ell$ exists for the $\CSS$-structures on tuples constructed in the proof of Lemma~\ref{lem:S}.
One also sees that in Lemma~\ref{lem:fg}, taking $\ell$ large enough ensures that the constructed polymorphism is also a~polymorphism of $(\CSS(\hH_i),\neq)$.
Thus, the model-complete cores of these structures, i.e., the structures $(\sS(\hH_1,g(1)),\neq)$, \dots, $(\sS(\hH_n,g(n)),\neq)$, also have a~quasi near unanimity polymorphism.
Moreover, these quasi near unanimity polymorphisms have the property that they do not identify any tuples other than those required to be identified by the quasi near unanimity identities.
Hence, since the superposition $\sS_n$ is generic, $(\sS_n,\neq)$ has a~quasi near unanimity polymorphism of arity $\ell$ as well.

By~\cite{BartoPinskerDichotomy, Topo}, it follows that $\Pol(\sS_n,\neq)$ has a~pseudo-Siggers operation for all $n\geq 1$.
Fix, for each $n\geq 1$, a~pseudo-Siggers operation $p_n\in \Pol(\sS_n,\neq)$. We can write
$$
\Pol(\sSa,\neq)=\bigcap_{n\geq 1} \Pol(\sS_n,\neq)\; .
$$
 By a~standard compactness argument, there exist $\alpha_n \in \Aut(\sSa)$ for all $n\geq 1$ such that the sequence $(\alpha_n\circ p_n)_{n\geq 1}$ converges pointwise to a~function $p$. Clearly, $p\in \Pol(\sSa,\neq)$.

We finish the proof by showing that $p$ is a~pseudo-Siggers polymorphism of $(\sSa,\neq)$. Let $F$ be a~finite subset of the domain of $\sSa$. Then on $F$ we have $p=\alpha_n\circ p_n$ for almost all $n\geq 1$. By the same argument as for $(\sSa,\neq)$, one sees that each $(\sS_n,\neq)$ is a~model-complete core.
Hence, since $p_n$ is a~pseudo-Siggers polymorphism of \mbox{$(\sS_n,\neq)$}, there exists $\beta_n\in\Aut(\sS_n)$ such that $p_n(x,y,x,z,y,z)=\beta_n\circ p_n(y,x,z,x,z,y)$ for all $x,y,z\in F$. Altogether, we get that for almost all $n\geq 1$ we have that for all $x,y,z\in F$
\begin{align*}
p(x,y,x,z,y,z)&=\alpha_n\circ p_n(x,y,x,z,y,z)\\
&=\alpha_n\circ \beta_n\circ p_n(y,x,z,x,z,y)\\
&=\alpha_n\circ \beta_n\circ
(\alpha_n)^{-1}\circ p(y,x,z,x,z,y)\; .
\end{align*}
This means that for almost all $n\geq 1$, there exists an automorphism of $\sS_n$ such that $p(x,y,x,z,y,z)$ can be composed with that automorphism from the outside to obtain $p(y,x,z,x,z,y)$ on $F$. By a~standard compactness argument, there exists an automorphism of $\sSa$ with this property. Again by a~standard compactness argument, there exist endomorphisms of $\sSa$ witnessing that $p$ is a~pseudo-Siggers polymorphism of $(\sSa,\neq)$.
\end{proof}

We can therefore apply the following result from~\cite{BKOPP17}.

\begin{theorem}\label{thm:equations}
  Let ${\mathscr C}$ be the polymorphism clone of an $\omega$-categorical model-complete core. Suppose that
  \begin{itemize}
    \item ${\mathscr C}$ satisfies a~non-trivial height 1 identity modulo outer unary functions, and
    \item ${\mathscr C}$ has a~uniformly continuous minion homomorphism to~$\Proj$.
  \end{itemize}
  Then ${\mathscr C}$ has at least doubly exponential orbit growth.
\end{theorem}

\begin{lemma}\label{lem:localh1}
\changed{Suppose that $\sSa$ has less than doubly exponential orbit growth.} Then  there is no uniformly continuous minion homomorphism from $\Pol(\sSa,\neq)$ to $\Proj$.
\end{lemma}
\begin{proof}
\changed{By Lemma~\ref{lem:psiggers}, $\Pol(\sSa,\neq)$ contains a~pseudo-Siggers operation, and by Lemma~\ref{lem:core}, $(\sSa,\neq)$ is an $\omega$-categorical model-complete core.} Hence, Theorem~\ref{thm:equations} implies that $\Pol(\sSa,\neq)$ has no uniformly continuous minion homomorphism to $\Proj$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main-3}]
{Let $g \colon {\mathbb N} \to {\mathbb N}$
be the function from {the last statement of} Lemma~\ref{lem:orbitgrowth}.
The structure $(\sSa,\neq)$ from  Construction~\ref{constr:s} is an $\omega$-categorical model-complete core without algebraicity (Lemma~\ref{lem:core})
and has less than doubly exponential orbit growth  (Lemma~\ref{lem:orbitgrowth})}. Moreover, $\Pol(\sSa,\neq)$ has a~minion homomorphism to $\Proj$ by Lemma~\ref{lem:noh1}, but no uniformly continuous such map by Lemma~\ref{lem:localh1}.
\end{proof}

config: arxiv


TITLE: Basis smaller than a topology
QUESTION [0 upvotes]: If $X=\{1,2,3\}$ and $\tau= \{X, \phi, \{1\}\}$, is there a basis for this that is not the topology itself?

REPLY [2 votes]: In any space $\mathcal{B}:=\mathcal{T}\setminus \{\emptyset\}$ is a base unequal to the topology. Also here.
$\{\{1\}\}$ is a subbase for the topology (the topology is the smallest one that contains this subbase).


TITLE: Update to SVD by changing 2 row vectors
QUESTION [2 upvotes]: Suppose I have a matrix in the form
$\begin{bmatrix}
a_1 \\
B \\
c_1
\end{bmatrix}
$
(The blocks of this matrix should be vertically stacked...don't know why the latex is wrong)
and its SVD is X.  Is there an efficient way of computing the SVD of
$
\begin{bmatrix}
a_2 \\
B \\
c_2
\end{bmatrix}
$
Where I replace the first row vector and the last row vector with new values, but keep B?

REPLY [1 votes]: From Golub and Van Loan, Matrix Computations 3rd Edition, Section 12.5 ("updating the ULV decomposition":

...$O(n^3)$ flops are required to recompute the SVD of a matrix that has undergone a unit rank perturbation.

If you just need the nullspace, you can get away with the ULV decomposition (L=lower triangular, U and V as in SVD), otherwise I am afraid you are out of luck.
But, as usual, the next question is: what do you need the SVD for? Do you need only the larger/smaller singular triples, or the whole decomposition?


TITLE: What's the difference between an orbital and an energy level? Are they the same thing?
QUESTION [1 upvotes]: In chemistry, you're likely to learn that each atom has a given number of orbitals, or electron shells, where the first orbital contains two electrons, the next contains eight electrons, and so on. In physics, you're likely to learn about electron excitation (electrons going from a lower energy level to a higher one) and photon emission (electrons returning from higher energy levels to lower energy levels). The concepts of orbitals and energy levels are clearly related - with each atom having a predetermined number of each - but the number of orbitals of an atom doesn't always (or ever, I'm not sure) equal the number of energy levels of that same atom.
For example; hydrogen has one orbital (containing one electron) but has 5 discrete energy levels at which the electron can exist.
If anyone could clear up my confusion it would be much appreciated. Thanks in advance!

REPLY [2 votes]: For each system you look at (e.g. atoms or molecules) there is a whole spectrum of a infinite number of orbitals and energy levels. From a theoreticians point of view, you can put in the electrons into these orbitals as you like to get a specific state of the system. With each occupied orbital you occupy the corresponding energy level. But of course you cannot occupy all oritals/levels due to the limited number of electron in a system. So to be precise:

each (physical) system has an infinity number of energy levels
each orbital has a energy level
there are occupied and unoccupied orbitals
the layout (which orbitals are occupied by electrons) determines the state of the system
the hydrogen atom has a infinite number of "occupieable" orbitals, but only one is occupied.

The chemists calssify the orbitals of single ions by labeling 1s 2s 2p ... 3d .... But these are classes of orbitals not the orbitals itself. For example in the class of the 3d orbitals every orbitals has a angular momentum quantum number l=2 and thus the class yields orbitals with angular-z-momentum m=-2,-1,0,1,2 and electrons of spin up and down, thus 10 orbitals.
But be aware that the thinking of orbitals and energy levels in just an approximation. The true quantum mechanical state of a system is given by its wave function, which is far more complex and yield much more physical effects that can be described by a simple orbital picture.


TITLE: Green function for interacting system
QUESTION [0 upvotes]: If we can diagonalize our interacting Hamiltonian then can we write a Green's function like we do for a non-interacting system? Green's function here means Matsubara in frequency-momentum space,
  $G(w,k)=w-E(k)$.

REPLY [1 votes]: yes you can write the green function in $G(w,k)=w-E(k)$ form if you have diagnolized your hamiltonian


TITLE: arg min_X ||A X B - C||^2, with X diagonal
QUESTION [1 upvotes]: Let $A, B, C$ be known matrices, and let $X$ be an unknown matrix.
Given that $C = AXB \Leftrightarrow \text{vec}(C) = K \text{vec}(X)$, where $\text{vec}(\cdot)$ denotes the vectorization of a matrix, $K = (B^{T} \otimes A)$ and $\otimes$ is the Kronecker product, it is easy to infer the least squares solution for $X$:
$$\arg\min_{X} || A X B - C ||^{2} = (K^{T} K)^{-1} K^{T} \text{vec}(C).$$
Is there a closed form solution also for the case in which $X$ is a diagonal matrix, i.e. for the following problem?
$$\arg\min_{X = diag(x_{1}, \ldots, x_{n})} || A X B - C ||^{2}.$$
Thank you guys in advance for your time :)

REPLY [1 votes]: Yes, it is possible. Notice that $AXB$ is a linear function in each $x_i$, so you can write it as $Mx$, for some $M\in\mathbb{C}^{n^2\times n}$. With a little work one can find out that the $i$th column of $M$ is given by the Kronecker product of the $i$th column of $A$ and the $i$th row of $B$, so one can write this matrix explicitly (if needed). If I am not mistaken, this can be written as $(B\otimes \mathbf{1})\circ(\mathbf{1}\otimes A)$, where $\otimes$ is the Kronecker product, $\circ$ is the Hadamard product, and $\mathbf{1}\in\mathbb{R}^{n}$ is the vector of all ones.
So you can write your problem as the standard least-squares problem
$$
\arg \min_{x_1,x_2,\dots, x_n} ||Mx-\operatorname{vec}(C)||^2,
$$
which is even easier to solve in practice than the original one, since $M$ has a small dimension $n$. A closed form solution is $(M^TM)^{-1}M^T\operatorname{vec}(C)$.