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Enter the Ring Edition by D. A. Horton; Elicia Horton and Publisher NavPress. Save up to 80% by choosing the eTextbook option for ISBN: 9781631466977, 1631466976. The print version of this textbook is ISBN: 9781631466953, 163146695X.


text:
added on September 18, 2017 Updated on September 18, 2017
We recently received a donation of this set of framed photographs of several boats which carried the inscription: Presented to P. Edwards, Esq., by the boys of the Concord Dinghi Club.
We have been unable to find any information of this club or of the gentleman named. Can you help, please?
Shackleton's Shipwreck: A Story of Survival
Twenty-seven people were all there were when Shackleton's Antarctic expedition went awry. Luckily for us (they were lucky enough, all surviving after months and months), one of those was a photographer, Frank Hurley, whose amazing photographs documented the 22 months spent stranded on the world's most remote continent. Ernest Shackleton set out in 1914 for…
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Russell Lea Manor
Russell Lea Manor, also known as Russell Lea House, was the home of Russell Barton (1830-1916), and was situated north of Lyons Road between Sibbick Street and Lyons Road. The suburb of Russell Lea takes its name from this grand home. Russell Barton rose from humble beginnings to become a pastoralist, mine-owner and politician. In the late 1870s…
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The Scottish Rifles – Early Beginnings
Bysiteadmin@michel.com.au December 14, 2017
The first NSW Military Unit with a Scottish Association was The Duke of Edinburgh's Highlanders formed in 1868. The Unit wore a kilt of Black Watch tartan and became known as the Highland Brigade before it was disbanded in 1878, following a decline in numbers caused by curtailment of new enrolment by Sir Henry Parkes, then the Premier of…
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Five Dock – 1888
The borough of Five Dock, which includes Burketown, Drummoyne, and Birkenhead, was incorporated on the 25th of July, 1871. It is bounded on the north by the Parramatta River, on the east by Iron Cove Creek and Parramatta-road, on the west by Concord and Hen and Chicken Bay, and on the south by Ashfield. lt…
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From the Magic Lantern to Television
When the 20th century was new, Australians took active part in most of their amusements. At home they would play musical instruments, or they would play cards or other games, they would watch magic lantern shows, or they would read. If they went out it was likely to be to a dance, unless it was…
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The Wreck of SS Admella, 1859
Bysiteadmin@michel.com.au January 23, 2018
Why is the wreck of this ship being remembered in the Australian maritime history books all these years later? Because it remains one of the worst maritime disasters in Australian history resulting in the tragic loss of 89 lives, mostly due to cold and exposure. They were in sight of land but unable to reach…
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Clare Stapleton says:
Yes! I will send you the article. Can you tell me if the inscription below the central photo says "State Champion 1921-1922" Clare
CCBHS Editor says:
The date shown in 1921-1922
I have now written a booklet on the Six foot sailing dingy period. I would like to get a better image of the plaque to use as the cover. Can you contact me to arrange? Clare
Kate Foot says:
It is possible that they are early versions of the 12-foot racing skiffs.
I know nothing more than that.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
The liquid crystalline phase is an intermediate state of matter between solid crystal and isotropic liquid, possessing orientational ordering but not positional \cite{DeGennes,Virga}. There are three major classes of liquid crystals, we shall consider the nematic phase only; liquid crystals in the nematic phase are aggregates of rod-like molecules whose preferred direction of alignment is parallel to its neighbour molecules. One of the key features of liquid crystals are point defects, which are points of discontinuity in the preferred direction of the liquid crystal, they can occur naturally in a system either by the geometry of the domain \cite{Boundary_defects_Boojums} or by boundary conditions \cite{Apala_bistable}. We shall be consider a thin film of nematic liquid crystal coating a curved surface, this is referred to as a nematic shell \cite{Nematic_Shells}. Nematic shells have a wide variety of technological applications ranging from catalysis, photonic band gap materials and creation of colloidal materials using DNA strands \cite{Tetravalent_Chemistry}, however the most novel application is to contact lenses that can change their refractive index and dioptre \cite{LC_Contact_Lenses}.

This piece can be viewed in two parts, the first part is the derivation of a conservation law for defects in a nematic shell, this shall be similar to the Poincaré-Hopf theorem however we shall consider domains with boundaries and their impact on the sum total of defects. The second part is investigating the bulk energy of defects in a nematic shell, as we shall be considering an energy density similar to the one constant approximation of the Frank free energy \cite{Frank} the presence of non-zero defects shall result in a divergence in the energy density. This allows us to consider local energies about defects and minimise the rate of divergence with respect to the defect strengths, thus allowing us to predict what families of defects shall appear.
\section{Topology Preamble}
	\subsection{Surface assumptions.}\label{Surface assumptions}
        Consider $S \subset \mathbb{R}^3$, an orientable two-dimensional manifold embedded in three dimensions, which is such that there exists an open, connected and bounded set $\Omega \subset \mathbb{R}^2$, and a single smooth chart $\mathbf{x}^{-1}: S\rightarrow \Omega$ with inverse $\mathbf{x} \in C^2(\overline{\Omega}, \overline{S})$. The regularity of the boundary $\partial S$ shall be discussed in the next section. We can form a "natural trihedron", a set of local basis vectors $\{\mathbf{e}^1,\mathbf{e}^2, \mathbf{N}\}$ relative to the surface
    	\begin{equation}\nonumber
    	    \mathbf{x}^i := \frac{\partial \mathbf{x}}{\partial \omega_i}, \quad \mathbf{e}^i := \frac{\mathbf{x}^i}{|\mathbf{x}^i|}, \quad
    	    \mathbf{N}:= \frac{\mathbf{x}^1\times \mathbf{x}^2}{\left| \mathbf{x}^1\times \mathbf{x}^2 \right|}, \text{ for } \underline{\omega}:=(\omega_1,\omega_2) \in \Omega.
        \end{equation}
        \begin{figure}[hbtp]
        \centering
        \includegraphics[width = 0.5 \textwidth]{Surface.png}
        \end{figure}
        We shall assume that the parametersiation of the surface $\mathbf{x}$ is such that the natural trihedron forms an orthogonal basis $\mathbf{x}^1 \cdot \mathbf{x}^2 (\underline{\omega}) = 0$ for all $\underline{\omega} \in \Omega$. The function
        \begin{equation}\label{Assumption on the Christoffel symbols}
            \mathbf{F}:= \left(\begin{array}{r}
        	-\frac{1}{|\mathbf{x}^2|}\frac{\partial |\mathbf{x}^1|}{\partial \omega_2} \\
        	\frac{1}{|\mathbf{x}^1|}\frac{\partial |\mathbf{x}^2|}{\partial \omega_1}
        	\end{array} \right)
        \end{equation}
    	is such that $\frac{\partial F_2}{\partial \omega_1} - \frac{\partial F_1}{\partial \omega_2} \in L^1(\Omega)$, this is a function of the first fundamental form's and is important in defining concepts such as the Gaussian Curvature. A notable example would be if $S$ is a surface of revolution, for a sufficiently smooth cross-section.
    \subsection{Boundary assumptions.}\label{Boundary assumptions}
        The Poincaré-Hopf theorem \cite{Differential_Geo}, is a conservation law for tangential vector fields on a closed surface (such as a torus or a sphere) or for tangential fields with boundary data equal to the outward pointing normal of the boundary to the surface. We seek to generalise this principle for surfaces with piecewise smooth boundaries and for vector fields with sufficiently regular boundary conditions. In this section we seek to parameterise the boundary $\partial S$, of the surface $S$, and describe its regularity.

        Assume that the boundary $\partial S \subset \mathbb{R}^3$ can be partitioned into a finite number of closed components, denoted $\partial S_1, \ldots, \partial S_M$ for $M \in \mathbb{N}$. We shall ignore the case when $M=0$, as there is no boundary (a torus or a sphere are classic examples) and thus the Poincaré-Hopf theorem may be applied.

        We parameterise each boundary component $\partial S_i$ by the function $\underline{\gamma}_i \in C([0,l_i],\partial S_i)$ where:
        \begin{itemize}
\item $l_i>0$ is the arclength of the curve $\partial S_i$ and the function $\underline{\gamma}_i$ is parameterised by its arclength, consequently $|\underline{\gamma}_i'|=1$.
\item $\underline{\gamma}_i(0)=\underline{\gamma}_i(l_i)$ because $\partial S_i$ is a closed curve.
\item There exist a finite number of values $0=:l^0_i< \ldots< l^n_i:=l_{i}$ where the curve is not twice differentiable: $$\underline{\gamma}_i \in \bigcup\limits_{j=1}^{n_i}C^2((l^{j-1}_i,l^j_i),\partial S_i), $$
these are called the \textit{verticies} of $\partial S_i$.
        \end{itemize}
        We notice that the parameterisation is twice differentiable on closed intervals that implies that the Geodesic curvature is integrable: \begin{equation}\nonumber
            kg_i := \underline{\gamma}_i''\cdot \left(\left(\mathbf{N}\circ \mathbf{x}^{-1}\circ\underline{\gamma}_i\right)\times \underline{\gamma}_i'\right) \in L^1((0,l_i)).
        \end{equation}

    The function $\tau^S_i: [0,l_i) \rightarrow (-\pi,\pi)$ describes the exterior angle of the boundary $\partial \Omega$, and is defined point-wise by
        \begin{equation}\nonumber
            \tau^S_i(s) := \lim\limits_{\delta \rightarrow 0}\left\{\begin{array}{ll}
                \text{sign}\left({det}(B^{i}(s,\delta))\right) |\arccos(\underline{\gamma}_i'(s-\delta) \cdot \underline{\gamma}_i'(s+\delta))| & \text{if } s\in (0,l_i) \\
                & \\
                \text{sign}\left({det}(B^{i}(s,\delta))\right) |\arccos(\underline{\gamma}_i'(l_i-\delta) \cdot \underline{\gamma}_i'(\delta))| & \text{if }s = \{0\}
            \end{array} \right.
        \end{equation}
        where the matrix
        \begin{equation}\nonumber
            B^{i}(s,\delta) := \left\{ \begin{array}{ll}
                \left(\underline{\gamma}_i'(s-\delta), \underline{\gamma}_i'(s+\delta), \frac{\underline{\gamma}_i'(s-\delta) \times \underline{\gamma}_i'(s+\delta)}{|\underline{\gamma}_i'(s-\delta) \times \underline{\gamma}_i'(s+\delta)|} \right) & \text{if }s \in (0,l_i) \\
                & \\
                \left(\underline{\gamma}_i'(l_i-\delta), \underline{\gamma}_i'(\delta), \frac{\underline{\gamma}_i'(l_i-\delta) \times \underline{\gamma}_i'(\delta)}{|\underline{\gamma}_i'(l_i-\delta) \times \underline{\gamma}_i'(\delta)|} \right)  & \text{if }s = \{0\}
            \end{array}\right.
        \end{equation}
        From our earlier assumptions about the continuity of $\underline{\gamma}_i$ the set $\text{supp} (\tau_i^S)$ is finite. These points are the vertices of $\partial S$, for $i=1,\ldots, M$  let the set $\{\tau_k\}_{k=1}^{\rm{V}_{\partial S}}$ be the set of exterior angles of $\partial S$ given by:
        \begin{equation}\nonumber
        	\{\tau_k\}_{k=1}^{\rm{V}_{\partial S}} := \bigcup_{i=1}^{M} \left\{\left.\tau_i^S(s)\right| \ s \in \text{supp}(\tau_i^S)\right\},
        \end{equation}
        for $\rm{V}_{\partial S} \in \mathbb{N}_0$.
    \subsection{Triangulation Definition.}
	    \begin{definition}
            An open subregion $S_R\subset S$ is simple if it is homeomorphic to a disk and its boundary $\partial S_R$ is parameterised by $\underline{\gamma}:[0,l]\rightarrow \partial S_R$ which satisfies the assumptions from section \ref{Boundary assumptions}. The simple subregion $S_R \subset S$ is a triangle if the corresponding exterior angle function $\tau:[0,l)\rightarrow  (-\pi,\pi)$ is such that $|\text{supp}(\tau)| = 3$. For a triangle $T \subset S$, let $\{s_1,s_2,s_3\} = \text{supp}(\tau)$ be such that $s_1<s_2<s_3$. We define the set of vertices $\mathcal{V}_T\subset \partial T $ and the set of edges $\mathcal{E}_T \subset \partial T$ by:
            \begin{equation}\label{Triangulation Verticies and Edges definition}
            	\mathcal{V}_T:= \left\{\underline{\gamma}(s_i)\right\}_{i=1}^3, \ \mathcal{E}_T:= \left\{\left\{\left.\underline{\gamma}(s)\right|s \in (0,s_1)\cup (s_3,l)\right\},\left\{\left.\underline{\gamma}(s)\right|s \in (s_1,s_2)\right\},\left\{\left.\underline{\gamma}(s)\right|s \in (s_2,s_3)\right\}\right\}
            \end{equation}
        \end{definition}
        In this piece we derive our main result by considering partitioning the domain into simple triangular subregions, we then derive a smaller result for each simple region and use that to derive the main result. The simple subregions will be important in the integration of various curvatures and angles, additionally they are used in the definition of a particular partition, known as a Triangulation of the surface, those familiar with finite element methods would have an illustration of this principle.
        \begin{definition}\label{Triangulation of a surface}
            Let us assume that $S$ is such that there exists a finite partition $\{T_i\}_{i=1}^{\rm{F}}$ , for such that each face $T_i \subset S$ is a triangle, let $\rm{F}\in \mathbb{N}$ denote the number of faces, and that the edge of a triangle is not "split" by a vertex of another triangle:
            \begin{equation}\nonumber
            	\partial T_i \cap \partial T_j = \overline{\mathcal{E}_{T_i} \cap \mathcal{E}_{T_j}}, \quad \forall i \neq j.
            \end{equation}

            We denote the number of vertices of this triangulation by $\rm{V} := \left| \bigcup\limits_{i=1}^{\rm{F}} \mathcal{V}_{T_i}\right|$ and the number of edges similarly $\rm{E}:=\left|\bigcup\limits_{i=1}^{\rm{F}} \mathcal{E}_{T_i}\right|$. For a given triangulation of $S$, the number
            \begin{equation}\nonumber
                \chi := \rm{F} - \rm{E} + \rm{V},
            \end{equation}
            is called the Euler characteristic of the triangulation. It is important to note that the Euler characteristic is invariant with respect to the choice of triangulation and thus is referred to as the Euler characteristic of the surface $S$.
        \end{definition}
       We wish to define our admissible class of unit vector fields, which are tangential to the surface $S$ permit the existence of "point defects" but no defects of any other type, and can satisfy the Poincaré–Hopf theorem.
        \begin{definition}\label{Chapter 2: Admissible space}
            We say that $\mathbf{u}:S \rightarrow \mathbb{S}^2$ is of the class $\mathcal{A}_{tan}$ if and only if there exists a vector field $\mathbf{A}:\overline{S}\rightarrow \mathbb{R}^3$ whose components are real analytic with isolated zeros, such that
            \begin{equation}
            	\mathbf{A}(\mathbf{x}(\underline{\omega}))\cdot \mathbf{N}(\underline{\omega}) = 0, \ \forall \underline{\omega} \in \Omega, \quad \mathbf{u}:= \frac{\mathbf{A}}{|\mathbf{A}|}, \ \text{ a.e }S.
            \end{equation}
        \end{definition}
        The choice of isolated zeros, of the corresponding analytic function, implies that there exists a discrete finite set $J_{\mathbf{u}} \subset \overline{S}$ such that $\mathbf{u} \in C^{\infty}(\overline{S}\setminus J_{\mathbf{u}}, \mathbb{S}^2)$, which is the set of discontinuities of the vector field $\mathbf{u}$ and corresponds to the isolated zeros of $\mathbf{A}$:
            \begin{equation}\label{Definition of Jump Set}
            	J_{\mathbf{u}}:= \left\{ \underline{s}\in \overline{S}\left| \mathbf{A}(\underline{s}) = \underline{0} \right. \right\}.
            \end{equation}
        	The parameterisation $\mathbf{x}$ was assumed to be bijective, as such there exists a $\mathbf{x}^{-1}\in C^{\infty}(\overline{S}, \Omega)$ such that $\mathbf{x}\circ \mathbf{x}^{-1}  = \text{Id}_{\Omega}$ and $\mathbf{x}^{-1}\circ \mathbf{x} = \text{Id}_{\overline{S}}$, we use this to construct the following decomposition. As $\mathbf{u}\in \mathcal{A}_{tan}$ is orthogonal to the normal $\mathbf{N}$, we may construct the following decomposition, there exists $(u_1,u_2) \in C^{\infty}(\overline{S}\setminus J_{\mathbf{u}}, \mathbb{S}^1)$ such that
            \begin{equation}\label{decomposition of a tangential field}
            	\mathbf{u}(\underline{s}) = u_1(\underline{s})\mathbf{e}^1(\mathbf{x}^{-1}(s)) + u_2(s)\mathbf{e}^2(\mathbf{x}^{-1}(s)), \quad \forall \underline{s} \in \overline{S}\setminus J_{\mathbf{u}}.
            \end{equation}

	\subsection{Curvature definitions.}
As we are considering a vector field on a curved surface, this motivates the question "how do we differentiate with respect to curved space?", the answer is the covariant derivative and Christoffel symbols. For $i,j,k = 1,2$, we define the Christoffel symbols of the second kind $\Gamma_{ij}^k$ and the functions of the second fundamental form $L_{ij}$, by the following:
        \begin{equation}\nonumber
			\mathbf{x}^{ij}:= \frac{\partial^2 \mathbf{x}}{\partial \omega_i \partial \omega_j} = \Gamma_{ij}^k\mathbf{x}^{k} + L_{ij}\mathbf{N}, \quad \Gamma_{ij}^k:=\frac{\mathbf{x}^{ij}\cdot \mathbf{x}^k}{|\mathbf{x}^k|^2}, \quad L_{ij}:=\mathbf{x}^{ij}\cdot \mathbf{N}, \quad \Gamma^2_{1j}\frac{|\mathbf{x}^2|}{|\mathbf{x}^1|} = F_j.
        \end{equation}
        as a direct consequence we have that
        \begin{equation}\nonumber
        	\frac{\partial \mathbf{e}^1 }{\partial \omega_j} = F_j\mathbf{e}^2 + \frac{L_{1j}}{|\mathbf{x}^1|}\mathbf{N}, \quad \frac{\partial \mathbf{e}^2 }{\partial \omega_j} =-F_j\mathbf{e}^1 + \frac{L_{2j}}{|\mathbf{x}^2|}\mathbf{N}
        \end{equation}
        Consider a curve $\underline{\gamma} \in C^1([0,L],\overline{S})$ parametrised by its arclength, we wish to consider the derivative $\frac{\partial }{\partial t}\left(\mathbf{u}\circ \underline{\gamma}(t)\right)$ and understand how it affected by the curvatures $\Gamma^k_{ij}$ and $L_{ij}$. We shall establish some notation before we proceed, recall that we assumed that $\mathbf{x}$ is bijective, thus there exists a curve $\underline{\beta} \in C^1([0,L], \Omega)$ such that $\underline{\gamma} = \mathbf{x}\circ \underline{\beta}$; additionally we denote the following restrictions $\hat{\mathbf{u}}:= \mathbf{u}\circ \underline{\gamma}$, $\hat{u}_i:= u_i \circ \underline{\gamma}$, $\hat{\mathbf{e}}^i := \mathbf{e}^i\circ \underline{\beta}$, $\hat{\mathbf{N}}:= \mathbf{N}\circ \underline{\beta}$, $\hat{\mathbf{F}}:= \mathbf{F}\circ \underline{\beta}$ and $\hat{A}_{ij} :=\frac{L_{ij}}{|\mathbf{x}^i|}\circ \underline{\beta}$.

        For a given $\mathbf{u} \in \mathcal{A}_{tan}$ with components $(u_1,u_2) \in C^{\infty}(\overline{S}\setminus J_{\mathbf{u}}, \mathbb{S}^1)$ the derivative is given by
        \begin{equation}\nonumber
        	\frac{d \hat{\mathbf{u}}}{dt} = \left(\frac{d\hat{u}_1}{dt} - \hat{u}_2\underline{\beta}'\cdot \hat{\mathbf{F}}\right)\hat{\mathbf{e}}^1 + \left(\frac{d\hat{u}_2}{dt} + \hat{u}_1\underline{\beta}'\cdot \hat{\mathbf{F}}\right)\hat{\mathbf{e}}^2 + \sum\limits_{i,j=1}^2\left(\hat{u}_iA_{ij}\beta'_j\right)\hat{\mathbf{N}}.
        \end{equation}
        In this piece we wish to consider derivatives with respect to the curved surface, as our vector field is tangent and does not have a normal component, this motivates us to consider a derivative with the same property. Consequently we define the co-variant derivative as the projection of the derivative into the tangent plane
        \begin{equation}\nonumber
        	\frac{D\hat{\mathbf{u}}}{dt} := \frac{d \hat{\mathbf{u}}}{dt} - \left( \frac{d \hat{\mathbf{u}}}{dt}\cdot \hat{\mathbf{N}}\right) \hat{\mathbf{N}} = \left(\frac{d\hat{u}_1}{dt} - \hat{u}_2\underline{\beta}'\cdot \hat{\mathbf{F}}\right)\hat{\mathbf{e}}^1 + \left(\frac{d\hat{u}_2}{dt} + \hat{u}_1\underline{\beta}'\cdot \hat{\mathbf{F}}\right)\hat{\mathbf{e}}^2
        \end{equation}
          \begin{figure}[htbp]
          \centering
          \includegraphics[width = 0.5\textwidth]{Covariant_Derivative.png}
          \end{figure}

          As $\hat{\mathbf{u}}$ is both unit length and tangential to the surface we have that $\frac{D\hat{\mathbf{u}}}{dt}$ is orthogonal to both $\hat{\mathbf{u}}$ and $\hat{\mathbf{N}}$. Thus there exists a real function $\left[\frac{D\mathbf{u}}{dt} \right]:[0,L] \rightarrow \mathbb{R}$ such that
        \begin{equation}\label{Definition of Algebraic Value}
            \frac{D\hat{\mathbf{u}}}{dt} = \left[\frac{D\mathbf{u}}{dt} \right]\left(\hat{\mathbf{N}}\times \hat{\mathbf{u}}\right), \quad \left[\frac{D\mathbf{u}}{dt} \right] := \frac{D\hat{\mathbf{u}}}{dt} \cdot \left(\hat{\mathbf{N}}\times \hat{\mathbf{u}}\right).
        \end{equation}
        This function is known as the Algebraic value, and can be used to  define important curvatures. One such curvature is the "Geodesic curvature" denoted $kg:[0,L]\rightarrow \mathbb{R}$, which is the Algebraic value of the vector $\underline{\gamma}'$, which shall be used later in the derivation of the main result. Additionally, the Algebraic value can be used to define the angle between two unit tangent vectors. For $\mathbf{v}\in \mathcal{A}_{tan}$ let $\hat{\mathbf{v}}:= \mathbf{v}\circ \underline{\gamma}$, the Algebraic value can be used to define the derivative of the angle $\phi_{\mathbf{u},\mathbf{v}}:[0,1]\rightarrow \mathbb{R}$ from $\hat{\mathbf{u}}$ to $\hat{\mathbf{v}}$,    \begin{equation}\label{angle between two vector fields}
            \frac{d \phi_{\mathbf{u},\mathbf{v}}}{dt} = \left[\frac{D\mathbf{v}}{dt}\right]- \left[\frac{D\mathbf{u}}{dt}\right].
        \end{equation}
        As a direct consequence of the above and the orthogonality of our basis, the derivative of the angle from $\hat{\mathbf{u}}$ to $\hat{\mathbf{e}}^1$, denoted $\phi:(0,L)\rightarrow \mathbb{R}$ is given by
               \begin{equation}\label{Equation 1}
            \left[\frac{D\mathbf{u}}{dt} \right]= \underline{\beta}'\cdot \hat{\mathbf{F}} + \frac{d\phi}{dt},
        \end{equation}
        The function $\phi$ will be key in defining the index of a vector field on a manifold.
        \begin{lemma}
            If the parametrisation is orthogonal, the Gaussian Curvature, denoted $K:\Omega \rightarrow \mathbb{R}$, of the surface $S$ is given by
            \begin{equation}\label{Gaussian Curvature}
                K := -\frac{1}{|\mathbf{x}^1||\mathbf{x}^2|}\left(\frac{\partial F_2}{\partial \omega_1} - \frac{\partial F_1}{\partial \omega_2} \right)
            \end{equation}
        \end{lemma}
        \begin{proof}
            See \cite{Differential_Geo}.
        \end{proof}
    \subsection{Strength of a defect}
        We can characterise a discontinuity by its Index (or defect strength in the context of liquid crystals, winding number if the surface is planar or degree of the mapping).
        \begin{definition}\label{definition index in terms of angles}
            Let $S_R \subset S$ be simple and with sufficiently smooth, positively oriented boundary $\partial S_R$ with parameterisation $\underline{\gamma}:[0,L]\rightarrow \partial S_R$, such that $\frac{d\phi }{ds}$ is integrable in $(0,L)$, then the index of $\mathbf{u}$ in $S_R$ is defined to be
            \begin{equation}\nonumber
            	\text{Index}(\mathbf{u},S_R):= \frac{1}{2\pi}\int\limits_{0}^L\frac{d\phi}{ds}\ ds,
            \end{equation}
            where $\phi$ is the angle from $\hat{\mathbf{u}}$ to $\hat{\mathbf{e}}^1$.
        \end{definition}
         The choice in the parameterisation $\underline{\gamma}$ implies that $\partial S_R$ is such that $|\partial S_R\cap J_{\mathbf{u}}| = \{0,1\}$, where the set $J_{\mathbf{u}}$ is given in equation (\ref{Definition of Jump Set}), thus we can only have one boundary defect in $R$. However, a reader might wonder "what if $|\partial R \cap J_{\mathbf{u}}|\geqslant 2$?" or "what if I want to know the index of a vector field on a non-simple domain?". The answer to both of those can be answered by the homotopy properties of the index. Considering an exact cover of simple domains $S^1_R,S_R^2, \ldots, S_R^N \subset S$, which satisfy the boundary assumptions. The index of $\bigcup\limits_{i=1}^N S^i_R$ is given by \cite{Defect_Group_theory}
            \begin{equation}\nonumber
                \text{Index}\left(\mathbf{u},\bigcup\limits_{i=1}^N S^i_R\right) := \sum_{i=1}^N \text{Index}(\mathbf{u} , S^i_R).
            \end{equation}
        This notion of a partition of a domain into regular components will be essential in the proof of the main result. However, the use of Algebraic values is rather cumbersome to use as a definition of the index, thus we wish to express the index as an integral of the components $(u_1,u_2)$ of $\mathbf{u}$.

        \begin{lemma}\label{Surface index to 2D index}
        	Let $\mathbf{u}\in \mathcal{A}_{tan}$ have the decomposition given by equation (\ref{decomposition of a tangential field}) then we have that
            \begin{equation}\nonumber
            	\text{Index}(\mathbf{u}, S_R) = \frac{1}{2\pi}\oint\limits_{\partial S_R} u_1 du_2 - u_2 du_1.
            \end{equation}
        \end{lemma}
        \begin{proof}
           The definition of the Algebraic value from equation (\ref{Definition of Algebraic Value}) and the decomposition of $\mathbf{u}$ from equation (\ref{decomposition of a tangential field}) implies that
           \begin{equation}\nonumber
           	\left[\frac{D\mathbf{u}}{dt} \right] = \frac{d\hat{\mathbf{u}}}{dt}\cdot \left(\hat{\mathbf{N}}\times \hat{\mathbf{u}} \right) = \hat{u}_1\left( \frac{d \hat{u}_2}{dt}+\hat{u}_1\underline{\beta}' \cdot \hat{\mathbf{F}}\right) - \hat{u}_2\left(\frac{d\hat{u}_1}{dt} - \hat{u}_2 \underline{\beta}'\cdot \hat{\mathbf{F}}\right)
           \end{equation}
           As the components satisfy  $u_1^2 + u_2^2 = 1$ we have that
            \begin{equation}\nonumber
            	\left[\frac{D\mathbf{u}}{dt} \right] = \hat{u}_1\frac{d \hat{u}_2}{dt} - \hat{u}_2\frac{d\hat{u}_1}{dt} + \underline{\beta}'\cdot \hat{\mathbf{F}}
            \end{equation}
            Comparing with equation (\ref{Equation 1}) we immediately deduce that $\frac{d\phi}{dt}=\hat{u}_1\frac{d \hat{u}_2}{dt} - \hat{u}_2\frac{d\hat{u}_1}{dt}$, which when substituted into the definition of Index, from equation (\ref{definition index in terms of angles}) yields our result.
        \end{proof}
\section{Topological charge conservation law for a simple domain}
    \begin{lemma}
        For $S_R\subset S$ simple with sufficiently smooth, positively oriented boundary $\partial S_R$ with sufficiently smooth parameterisation $\underline{\gamma}:[0,L]\rightarrow \partial S_R$, and $\mathbf{u}\in \mathcal{A}_{tan}$, we have that
        \begin{equation}\label{Integral of Gaussian Curvature definition}
            \int\limits_{0}^L\left( \frac{d \phi}{dt} - \left[\frac{D\mathbf{u}}{dt} \right] \right) \ dt = \iint\limits_{S_R}K \ d\sigma,
        \end{equation}
        where $\phi$ is the angle between $\hat{\mathbf{u}}$ and $\hat{\mathbf{e}}^1$.
    \end{lemma}
    \begin{proof}
        Consider the integral of equation (\ref{Equation 1}) over the interval $(0,L)$
        \begin{equation}\nonumber
            \int\limits_{0}^L\left( \frac{d \phi}{dt} - \left[\frac{D\mathbf{u}}{dt} \right] \right) \ dt = -\int\limits_{0}^L\underline{\beta}'\cdot (\mathbf{F}\circ \underline{\beta}) dt.
        \end{equation}
        Recall the assumption that the transformation $\mathbf{x}$ is bijective and smooth, this implies that there exists a simple $R \subset \Omega$ with boundary parametrised by $\underline{\beta}$ such that $\mathbf{x}(R)=S_R$. As the domain $R$ is simple we can apply Green's theorem to the right hand side to deduce that
        \begin{equation}\nonumber
            \int\limits_{0}^L\left( \frac{d \phi}{dt} - \left[\frac{D\mathbf{u}}{dt} \right] \right) \ dt = -\iint\limits_{R}\left(\frac{\partial F_2}{\partial \omega_1}-\frac{\partial F_1}{\partial \omega_2} \right)d \underline{\omega}.
        \end{equation}
        We now applying the definition of Gaussian curvature for orthogonal parametrisation, equation (\ref{Gaussian Curvature}), we deduce that
        \begin{equation}\nonumber
            \int\limits_{0}^L\left( \frac{d \phi}{dt} - \left[\frac{D\mathbf{u}}{dt} \right] \right) \ dt = \iint\limits_{R}K |\mathbf{x}^1||\mathbf{x}^2|d \underline{\omega}.
        \end{equation}
        Finally applying the definition of integration with respect to curvilinear co-ordinates we have that $|\mathbf{x}^1||\mathbf{x}^2|  d\underline{\omega} = d\sigma$, giving our result.
    \end{proof}
    \begin{lemma}
        For a surface $S$, satisfying the assumptions from sections (\ref{Surface assumptions})-(\ref{Boundary assumptions}), the Gauss-Bonnet theorem states that
        \begin{equation}\label{Gauss-Bonnet theorem}
            2\pi \chi(S) = \oint\limits_{\partial S} kg \ ds + \iint\limits_S K \ d\sigma + \sum_{i=1}^{\rm{V}_{\partial S}} \tau_i.
        \end{equation}
    \end{lemma}
    Utilising these two lemmas, the Gauss-Bonnet theorem and the definition of boundary index, we can derive a conservation law for the interior defects on simple surfaces.
    \begin{theorem}
        Let $S_R\subset S$ be a simple domain with sufficiently smooth boundary $\partial S_R$ which is parametrised by a $\underline{\gamma}:[0,L]\rightarrow \partial S_R$ (which is parametrised by its arclength $L>0$), additionally we denote the exterior angles of $\partial S_R$ by $\tau_i \in (-\pi,\pi)$ for $i=1,\ldots V_{e,\partial S_R}$. For a $\mathbf{u}\in \mathcal{A}_{tan}$ we have that
        \begin{equation}\label{Simple domain Poincare Hopf generalisation}
            2\pi = \sum_{i=1}^{\rm{V}_{\partial S_R}} \tau_i + \oint\limits_{\partial S_R} \frac{d \theta}{dt}\ dt + 2\pi \text{Index}(\mathbf{u},S_R),
        \end{equation}
        where $\theta$ is the angle between $\hat{\mathbf{u}}$ and the unit tangent to the boundary $\underline{\gamma}'$.
    \end{theorem}
    \begin{proof}
        The definition of a simple domain implies that it is homeomorphic to a disk, consequently it has Euler characteristic $\chi =1$. Applying the Gauss-Bonnet theorem for $R$ we deduce that
        \begin{equation}\nonumber
            2\pi  = \sum_{i=1}^{\rm{V}_{\partial S_R}} \tau_i + \oint\limits_{\partial S_R} kg \ dt + \iint\limits_{S_R} K \ d\sigma.
        \end{equation}

        The boundary of $S_R$ being parametrised by its arclength implies that, $|\underline{\gamma}'|=1$ we can substituting in equation (\ref{Integral of Gaussian Curvature definition}) to deduce that
        \begin{equation}\nonumber
            2\pi  = \sum_{i=1}^{\rm{V}_{\partial S_R}} \tau_i + \int\limits_{0}^L \left( kg -  \left[\frac{D\mathbf{u}}{dt} \right] \right)\ dt + \int\limits_{0}^L\frac{d \phi}{dt} \ dt,
        \end{equation}
        where $\phi$ is the angle between $\hat{\mathbf{u}}$ and $\hat{\mathbf{e}}^1$. Firstly we apply  the definition of index from equation (\ref{definition index in terms of angles}) to  express $\int\limits_{0}^L\frac{d \phi}{dt} \ dt$ in terms of the Index. Secondly, we recall that the Geodesic curvature, $kg:(0,L)\rightarrow \mathbb{R}$, is defined to be the algebraic value of $\underline{\gamma}$. Finally we use equation (\ref{angle between two vector fields}), to express that the integral of the derivative of the angle between two vectors is given by the difference of their algebraic values, which obtains our result.
    \end{proof}

\section{Topological charge conservation law for a non-simple domain}
    In the previous section we derived a conservation law for a tangential vector field on a simple domain, we shall use this to generalise to any sufficiently smooth surface. We shall do this by partitioning the surface into regular components, then apply the previous result on each component before using that to deduce our main result. However, we shall be considering a very specific partition known as a \textit{triangulation}, readers who are familiar with numerical finite element methods would have a good visualisation.

    Consider a triangulation, see definition (\ref{Triangulation of a surface}), $\mathcal{T}:=\{T_i\}_{i=1}^{\rm{F}}$ of the surface S, where $\rm{F} \in \mathbb{N}$ the number of triangles. For each triangle $T_i$ let the edges be denoted $\mathcal{E}_{T_i}=\{e_{ij}\}_{j=1}^3$, the exterior angles $\{\tau_{ij}\}_{i=1}^3$, and the parametrisation of the boundary $\underline{\gamma}_i \in C([0,L_i],\partial T_i)$ is parametrised by its arclength and is smooth on the edges. Additionally for a given $\mathbf{u}\in \mathcal{A}_{tan}$ it is assumed that $|\partial T_i \cap J_{\mathbf{u}}|\leqslant 1$, and if $|\partial T_i \cap J_{\mathbf{u}}|=1$ then $\underline{\gamma}_i$ is such that $\underline{\gamma}_i(0) = \partial T_i \cap J_{\mathbf{u}} = \underline{\gamma}_i(L_i)$. As triangles are simple domains, we may apply equation (\ref{Simple domain Poincare Hopf generalisation}) to the vector field on each triangle, hence
    \begin{equation}\nonumber
        2\pi =  \sum_{j=1}^3\tau_{ij} + \sum_{j=1}^3\int\limits_{e_{ij}} \frac{d\theta_i}{dt}dt  + 2\pi \text{Index}(\mathbf{u},T_i),
    \end{equation}
    where $\theta_i$ is the angle between $\mathbf{u}$ and $\underline{\gamma}_i'$. Taking a summation of the above equation over all triangles $\{T_i\}_{i=1}^{\rm{F}}$, thus yielding
    \begin{equation}\label{Appendix: Faces summation}
        2\pi \rm{F} = \sum_{i=1}^{\rm{F}}\sum_{j=1}^3 \tau_{ij} + \sum_{i=1}^{\rm{F}}\sum_{j=1}^3 \int\limits_{e_{ij}} \frac{d\theta_i}{dt}dt + 2\pi \sum_{i=1}^{\rm{F}}\text{Index}(\mathbf{u},T_i).
    \end{equation}
    We shall now simplify these terms, consider a pair of triangles $T_{i}$ and $T_{k}$ such that there exists $n,m \in \{1,2,3\}$ such that $e_{in} = e_{km}$. As the boundary of each triangle is positively oriented we have that
    \begin{equation}\label{appendix: angle integral}
        \int\limits_{e_{in}} \frac{d\theta_{i}}{dt}dt = - \int\limits_{e_{km}} \frac{d\theta_{k}}{dt}dt. \quad \Rightarrow \sum_{i=1}^{\rm{F}}\sum_{j=1}^3 \int\limits_{e_{ij}} \frac{d\theta_i}{dt}dt = \int_{\partial S} \frac{d\theta}{dt}dt,
    \end{equation}
   where the function $\theta$ is the angle between the vector field $\mathbf{u}$ and $\underline{\gamma}'$, where $\underline{\gamma}:[0,L]\rightarrow \partial S$ is positively oriented and parametrises the boundary $\partial S$. We shall now simplify the sum of the exterior angles, using the following notation:
    \begin{align}
        \rm{E}_{\partial S} &= \text{ number of external edges of }\mathcal{T}\nonumber \\
         \rm{E}_{S} &= \text{ number of internal edges of }\mathcal{T} \text{ (ignoring repeated edges)}.\nonumber \\
        \rm{V}_{\partial \mathcal{T}} &= \text{ number of external vertices of }\mathcal{T}.\nonumber \\
         \rm{V}_{\partial S} &= \text{ number of vertices of }\partial S. \nonumber \\
        \rm{V}_{\partial \mathcal{T} / S} &= \text{ number of vertices of our triangulation which are not vertices of }\partial S. \nonumber \\
         \rm{V}_{\mathcal{T}} &= \text{ number of internal vertices of }\mathcal{T}.\nonumber \\
        \sigma_{i,j} &:= \pi - \tau_{ij}, \text{ the interior angles of the triangle }T_i \nonumber
    \end{align}
    It is clear that $\rm{V}_{\partial \mathcal{T}/S} + \rm{V}_{\partial S} = \rm{V}_{\partial \mathcal{T}}$ and that
    \begin{equation}\nonumber
        \sum_{i=1}^{\rm{F}}\sum_{j=1}^3 \tau_{ij} = \sum_{i=1}^{\rm{F}}\sum_{j=1}^3 (\pi - \sigma_{i,j}) = 3\pi \rm{F} - \sum_{i=1}^{\rm{F}}\sum_{j=1}^3 \sigma_{i,j}.
    \end{equation}
    It can be shown that because we are considering a triangulation we have that $3\rm{F} = 2\rm{E}_S + \rm{E}_{\partial S}$ and consequently
    \begin{equation}\label{appendix: angle sum intermediate}
        \sum_{i=1}^{\rm{F}}\sum_{j=1}^3 \tau_{ij} = 2\pi \rm{E}_S + \pi \rm{E}_{\partial S} - \sum_{i=1}^{\rm{F}}\sum_{j=1}^3 \sigma_{i,j}.
    \end{equation}
    If a vertex is internal, then the sum of internal angles at that vertex is $2\pi$, and if a vertex is external but not a vertex of $\partial S$, then the sum of internal angles is $\pi$. Recall that $\{\tau_k\}_{k=1}^{\rm{V}_{\partial S}}$ is the set of external angles of $\partial S$, we have that
    \begin{equation}\nonumber
        \sum_{i=1}^{\rm{F}}\sum_{j=1}^3 \sigma_{i,j} = 2\pi \rm{V}_{\mathcal{T}} + \pi \rm{V}_{\partial \mathcal{T}/ S}  + \sum_{k=1}^{\rm{V}_{\partial S}} (\pi-\tau_k).
    \end{equation}
    As the boundary curves are closed it is clear that $\rm{E}_{\partial S} = \rm{V}_{\partial \mathcal{T}}$, therefore we have $\rm{E}_{\partial S} = 2\rm{E}_{\partial S} - \rm{V}_{\partial \mathcal{T}}$ and hence
    \begin{equation}\label{appendix: sum of exterior angles tessaltion}
        \sum_{i=1}^F\sum_{j=1}^3 \tau_{ij} = 2\pi (\rm{E}_S+ \rm{E}_{\partial S}) - 2\pi (\rm{V}_{\mathcal{T}} + \rm{V}_{\partial \mathcal{T}}) + \sum_{k=1}^{\rm{V}_{\partial S}} \tau_k.
    \end{equation}
    Reusing the same notation from definition (\ref{Triangulation of a surface}) it is clear that $\rm{E} = \rm{E}_S+ \rm{E}_{\partial S}$ and $\rm{V} = \rm{V}_{\mathcal{T}}+ \rm{V}_{\partial\mathcal{T}}$. Consequently substituting equations (\ref{appendix: angle integral}) and (\ref{appendix: sum of exterior angles tessaltion}) into (\ref{Appendix: Faces summation}) we obtain
    \begin{equation}\label{Conservation law for defects}
        \chi = \sum_{k=1}^{\rm{V}_{\partial S}} \frac{\tau_k}{2\pi} + \frac{1}{2\pi}\oint\limits_{\partial S} \frac{d\theta}{ds}\ ds + \sum_{i=1}^{\rm{F}} \text{Index}(\mathbf{u},T_i)
    \end{equation}
    where $\chi$ is the Euler Characteristic of the surface $S$, given in definition (\ref{Triangulation of a surface}).
\section{Oseen-Frank energy rate minimisation}
 In the previous section we derived a conservation law for vector fields on a surface. We would normally wish to consider the Dirichlet integral of this vector field to find the minimising configurations, however we shall show that if there exists a point with non-zero index then the energy is infinite. Thus we either need to consider systems without defects, or we change our question.

 Rather than minimising the Dirichlet integral of the vector field, we can instead minimise the rate at which the Dirichlet integral diverges; since the "minimising" configurations would be the ones which are the slowest to diverge. To derive such a rate we shall require a few additional assumptions on the geometry of our surface.
    \subsection{Geometry assumptions}\label{section: Additional geometery assumptions}
        We assume that our parameterisation $\mathbf{x}$ is such that the vector field $\mathbf{F}$ is such that $|\mathbf{F}|^2|\mathbf{x}^1||\mathbf{x}^2| \in L^1(\Omega)$, this is equivalent to saying that $\mathbf{F}$ is in the class $L^2$ with respect to curvilinear co-ordinates. This assumption shall play a minor role in the construction of a lower bound. Similarly we recall the definition of the first fundamental forms $L_{ij}$ and assume that
        \begin{equation}\nonumber
        	|\mathbf{x}^1||\mathbf{x}^2|\sum\limits_{i,j=1}^2 \frac{L_{ij}}{|\mathbf{x}^i|}\frac{L_{ij}}{|\mathbf{x}^i|} \in L^1(\Omega).
        \end{equation}

        Additionally, for each point $\underline{\hat{\omega}}\in \overline{\Omega}$ we assume there exists a $\rho(\underline{\hat{\omega}})>0$, $\alpha(\underline{\hat{\omega}}) \in [0,\infty)$ and $h^-(\underline{\hat{\omega}})\leqslant h^+(\underline{\hat{\omega}})$ such that
        \begin{equation}\label{Holder approximation}
            0< |\mathbf{x}^i|(\underline{\hat{\omega}}) + h^-|\underline{\omega}-\underline{\hat{\omega}}|^\alpha \leqslant |\mathbf{x}^i|(\underline{\omega}) \leqslant |\mathbf{x}^i|(\underline{\hat{\omega}}) + h^+|\underline{\omega}-\underline{\hat{\omega}}|^\alpha,\quad  \underline{\omega} \in \overline{\Omega} \cap B_{\rho}(\underline{\hat{\omega}})\setminus \{\underline{\hat{\omega}}\},
        \end{equation}
        for $i=1,2$. This local Hölder approximation is very important in the derivation of the divergence rate, especially in the case when $|\mathbf{x}^1|(\underline{\hat{\omega}})=|\mathbf{x}^2|(\underline{\hat{\omega}}) =0$.
  \subsection{Vector field assumptions}
Similar to the previous section we require some additional assumptions on the vector field $\mathbf{u}\in \mathcal{A}_{tan}$. A point $\underline{s}$ is in $J_{\mathbf{u}}\setminus \partial S$ if and only if $\mathbf{x}^{-1}(\underline{s}) \notin \partial \Omega$, we formally understand this as "an internal defect of $S$ is an internal defect of $\Omega$". For a given  with discontinuity set $J_{\mathbf{u}}\subset S$ , which we shall define the constant $R > 0$ to be
\begin{equation}\nonumber
	2R :=\left\{\inf\limits_{\underline{\omega}\in \mathbf{x}^{-1}(J_{\mathbf{u}})}\left(\text{dist}(\underline{\omega}, \mathbf{x}^{-1}(J_{\mathbf{u}})\setminus \{\underline{\omega}\}) \right), \inf\limits_{\underline{\omega}\in \mathbf{x}^{-1}(J_{\mathbf{u}}\setminus \partial S)}\left(\text{dist}(\underline{\omega}, \partial \Omega)\right), \min\limits_{\underline{\omega}\in \mathbf{x}^{-1}(J_{\mathbf{u}})}\rho(\underline{\omega})\right\}
\end{equation}
Thus the constant $R >0$ is such that for $\underline{\omega}^1, \underline{\omega}^2 \in \mathbf{x}^{-1}(J_{\mathbf{u}})$ distinct, the simple manifolds $\mathbf{x}(B_{R}(\underline{\omega}^1)\cap \Omega)$ and $\mathbf{x}(B_R(\underline{\omega}^2)\cap \Omega)$ do not intersect. For $\epsilon < R$ we define the following domain
        \begin{equation}\nonumber
            \Omega_\epsilon := \Omega \setminus \bigcup\limits_{\underline{\omega} \in J_{\mathbf{u}}} \bar{B}_\epsilon(\underline{\omega}).
        \end{equation}
        We consider the one constant approximation to the Oseen-Frank energy, denoted $E_{\epsilon}:\mathcal{A}_{tan}\rightarrow [0,\infty)$, on a manifold which is the simplest strain energy \cite{Nematic_shells_one_constant},
        \begin{equation}\nonumber
            E_{\epsilon}[\mathbf{u}] := \int\limits_{\mathbf{x}(\Omega_\epsilon)}\left|\text{grad}(\mathbf{u})\right|^2 \ d\sigma = \int\limits_{\Omega_{\epsilon}}\left(\left|\frac{D\mathbf{u}}{\partial \omega_1} \right|^2 + \left| \frac{D\mathbf{u}}{\partial \omega_2}\right|^2 + \left|\mathcal{B}[\mathbf{u}] \right|^2 \right)|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega},
        \end{equation}
where the co-variant derivative $\frac{D}{\partial \omega_i}$ and shape operator $\mathcal{B}$ are defined by
        \begin{equation}\nonumber
        	\frac{D\mathbf{u}}{\partial \omega_i} := \frac{\partial (\mathbf{u}\circ \mathbf{x})}{\partial \omega_i} - \left(\frac{\partial (\mathbf{u}\circ \mathbf{x})}{\partial \omega_i} \cdot \mathbf{N} \right)\mathbf{N}, \quad \mathcal{B}_j[\mathbf{u}] := \sum\limits_{i=1}^2 (u_i \circ \mathbf{x})\frac{L_{ij}}{|\mathbf{x}^i|}.
        \end{equation}
        We shall omit the "$\circ \mathbf{x}$" for tractability as we shall be considering functions of $\Omega$ only. We have excluded a small region about each of the defects and wish to estimate the local rate of divergence about each discontinuity, thus we seek a lower bound on the energy functional. Our assumption about the regularity of $L_{ij}$ means that
        \begin{equation}\nonumber
        	E_{\epsilon}[\mathbf{u}] \geqslant \int\limits_{\Omega_{\epsilon}\setminus \overline{\Omega}_R}\left(\left|\frac{D\mathbf{u}}{\partial \omega_1} \right|^2 + \left| \frac{D\mathbf{u}}{\partial \omega_2}\right|^2 \right)|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} = \int\limits_{\Omega_{\epsilon}\setminus \overline{\Omega}_R}\left|u_1\nabla u_2 - u_2 \nabla u_1 + \mathbf{F} \right|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega},
        \end{equation}
        where $\nabla$ is the gradient operator in $\Omega$. For a given $q_1 \in (0,1)$, we can bound the integrand further by the following inequality
        \begin{equation}\nonumber
        	|u_1\nabla u_2 - u_2 \nabla u_1 + \mathbf{F}|^2 \geqslant q_1 |u_1 \nabla u_2 - u_2 \nabla u_1|^2 - q_2, \quad q_2 \geqslant \frac{q_1}{1-q_1}|\mathbf{F}|^2.
        \end{equation}
        In the formulation we assumed that $|\mathbf{F}|^2|\mathbf{x}^1||\mathbf{x}^2|$ is integrable, thus the $q_2$ term is negligible for this analysis. We need to estimate
        \begin{equation}\nonumber
        	\begin{split}
        		\int\limits_{\Omega_{\epsilon}\setminus \overline{\Omega}_R}\left|u_1\nabla u_2 - u_2 \nabla u_1 \right|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} = \sum\limits_{\hat{\underline{\omega}}\in J_{\mathbf{u}}} \int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})} |u_1 	\nabla u_2 - u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2| \ d \underline{\omega}
        	\end{split}
        \end{equation}
        where $A^R_{\epsilon}(\hat{\underline{\omega}}):=\left(B_{R}(\hat{\underline{\omega}})\setminus \overline{B}_{\epsilon}(\hat{\underline{\omega}})\right) \cap \Omega$. We shall a lower bound on the local Dirichlet integral of an internal defect and then use that to motivate a similar analysis of a boundary defect. The definition of $R$ is such that $\rm{Index}(\mathbf{u}, \Omega \cap B_r(\underline{\hat{\omega}}))$ is constant for all $r \leqslant R$, we denote the index by $n(\underline{\hat{\omega}}):=\rm{Index}(\mathbf{u}, \Omega \cap B_r(\underline{\hat{\omega}}))$, we shall omit $\underline{\hat{\omega}}$ during the derivation of the lower bound. Additionally we define the functions $m^+, m^- : J_{\mathbf{u}}\rightarrow [0,\infty)$ by
        \begin{equation}\nonumber
        	m^+(\underline{\hat{\omega}}) := \max\{|\mathbf{x}^1|(\underline{\hat{\omega}}), |\mathbf{x}^2|(\underline{\hat{\omega}})\}, \quad m^-(\underline{\hat{\omega}}) := \min\{|\mathbf{x}^1|(\underline{\hat{\omega}}), |\mathbf{x}^2|(\underline{\hat{\omega}})\},
        \end{equation}
        similarly we shall omit the $\underline{\hat{\omega}}$, it is clear by definition that $0 \leqslant m^- \leqslant m^+$. We consider two cases, when $m^+ \neq 0$ and $m^+ = 0$, in this second case we also have that $m^-=0$.
        \begin{lemma}
        	\begin{itemize}
\item In the case $m^+\neq 0$ we have that $\lim\limits_{\epsilon \rightarrow 0}\int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})}|u_1\nabla u_2-u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} $ diverges to infinity with rate $2\pi n^2 \left(\frac{m^-}{m^+}\right)^2 \log \frac{1}{\epsilon}$.
\item In the case $m^+ = 0$ we have that $\lim\limits_{\epsilon \rightarrow 0}\int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})}|u_1\nabla u_2-u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} $ diverges to infinity with rate $2\pi n^2 \left(\frac{h^-}{h^+}\right)^2 \log \frac{1}{\epsilon}$, where $h^+,h^-$ are the bounds from equation (\ref{Holder approximation}).
        	\end{itemize}
 	\end{lemma}
    \begin{proof}
    	As we are considering an annulus the most appropriate basis would be polar co-ordinates $(r,\vartheta)$ which motivates
        \begin{equation}\nonumber
        	\mathbf{x}^1 = \mathbf{x}^r \cos \vartheta - \frac{1}{r}\mathbf{x}^\vartheta \sin \vartheta, \quad \mathbf{x}^2 = \mathbf{x}^r \sin \vartheta + \frac{1}{r}\mathbf{x}^\vartheta \cos \vartheta.
        \end{equation}
        Considering the cross and dot product of $\mathbf{x}^1$ and $\mathbf{x}^2$ we deduce that
        \begin{equation}\nonumber
        	\mathbf{x}^1 \times \mathbf{x}^2  = \frac{1}{r}\mathbf{x}^r \times \mathbf{x}^{\vartheta}, \quad \mathbf{x}^r \cdot \mathbf{x}^{\vartheta}=0, \quad  |\mathbf{x}^1||\mathbf{x}^2|d \underline{\omega} = |\mathbf{x}^r||\mathbf{x}^\vartheta|d\vartheta dr.
        \end{equation}
      Thus we have that
        \begin{equation}\label{part 1 interior defect lower bound}
        	\int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})}|u_1\nabla u_2-u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} \geqslant \int\limits_\epsilon^R\int\limits_{-\pi}^\pi\left(u_1\frac{\partial u_2}{\partial \vartheta} - u_2 \frac{\partial u_1}{\partial \vartheta} \right)^2 |\mathbf{x}^r||\mathbf{x}^\vartheta|d\vartheta \frac{dr}{r^2}
        \end{equation}
    Our goal will be to substitute the defect strength to eliminate the $\vartheta$ integral, if we consider the curve $C^r :=\{\mathbf{x}(\hat{\underline{\omega}}+ r \mathbb{S}^1)\}$ parametrised by $\underline{\gamma}_r(\vartheta) := \mathbf{x}(\hat{\underline{\omega}}+ r (\cos \vartheta, \sin \vartheta))$ for $\vartheta \in [-\pi,\pi)$ we note that $\mathbf{x}^{\vartheta}=\underline{\gamma}_r'$. It is clear from the definition of index and the assumption of isolated zeros that
    \begin{equation}\nonumber
    	2\pi n = \oint_{C^r}u_1du_2 -u_2 du_1 = \int\limits_{-\pi}^{\pi} \left(u_1\frac{\partial u_2}{\partial \vartheta} - u_2 \frac{\partial u_1}{\partial \vartheta} \right) |\mathbf{x}^{\vartheta}|d\vartheta , \quad \forall r \in (0,R).
    \end{equation}
    Comparing the above to equation (\ref{part 1 interior defect lower bound}) it is clear that we must bound $|\mathbf{x}^r|$ from below by a function of $r$ only. Reverting back to $\mathbf{x}^1$ and $\mathbf{x}^2$ we have that
    \begin{equation}\label{lower bound on xr}
    	|\mathbf{x}^r| = r\sqrt{|\mathbf{x}^1|^2\cos^2 \vartheta + |\mathbf{x}^2|^2\sin^2 \vartheta} \geqslant r \min\limits_{i=1,2}|\mathbf{x}^i|, \quad \forall r \in [0,R], \vartheta \in [-\pi,\pi]
    \end{equation}
    Recall we assumed that $\mathbf{x}^1$ and $\mathbf{x}^2$ satisfy equation (\ref{Holder approximation}) this allows us construct the following bounds
    \begin{equation}\label{lower bound on xi}
    	0< h^-r^{\alpha}+m^-  \leqslant |\mathbf{x}^i|\leqslant h^+r^{\alpha}+m^+, \quad \text{ for }i=1,2.
    \end{equation}
  	Substituting the lower bound from equation (\ref{lower bound on xi}) into equation (\ref{lower bound on xr}) and then applying that bound to $|\mathbf{x}^r|$ in equation (\ref{part 1 interior defect lower bound}) yields
    \begin{equation}\label{part 2 interior defect lower bound}
        	\int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})}|u_1\nabla u_2-u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} \geqslant \int\limits_\epsilon^R\oint\limits_{C^r}\left(u_1\frac{\partial u_2}{\partial \vartheta} - u_2 \frac{\partial u_1}{\partial \vartheta} \right)^2 d\lambda \  (m^- + h^- r^\alpha)\frac{dr}{r}.
        \end{equation}
    where $d\lambda := |\mathbf{x}^\vartheta|d\vartheta$. Using the Cauchy-Schwartz inequality we deduce that
    \begin{equation}\label{Cauchy-Schwartz Full circle}
    	4\pi^2 n^2 = \left( \oint_{C^r}\left(u_1\frac{\partial u_2}{\partial \vartheta} - u_2 \frac{\partial u_1}{\partial \vartheta} \right) \ d\lambda\right)^2 \leqslant \oint_{C^r} 1^2 d\lambda \oint_{C^r}\left(u_1\frac{\partial u_2}{\partial \vartheta} - u_2 \frac{\partial u_1}{\partial \vartheta} \right)^2 d \lambda,
    \end{equation}
We wish to bound $\oint_{C^r}\left(u_1\frac{\partial u_2}{\partial \vartheta} - u_2 \frac{\partial u_1}{\partial \vartheta} \right)^2 d \lambda$ from below in terms of the index of the vector field $n$ and the parameter $r$, thus we must construct an upper bound on the arclength of $C^r$,
\begin{equation}\label{Bound on arclength Cr}
	(m^-+ h^-r^\alpha)|\mathbf{x}^{\vartheta}|\leqslant \frac{1}{r}|\mathbf{x}^r||\mathbf{x}^\vartheta| = |\mathbf{x}^1||\mathbf{x}^2| \leqslant (m^++h^+r^\alpha)^2, \quad  \oint_{C^r} 1^2 d\lambda = \int\limits_{-\pi}^{\pi}|\mathbf{x}^{\vartheta}|d\vartheta\leqslant 2\pi \frac{ (m^++h^+r^\alpha)^2}{m^-+h^-r^\alpha}.
\end{equation}
Thus substituting the lower bound from equation (\ref{Bound on arclength Cr}) into the Cauchy-Schwartz inequality from equation (\ref{Cauchy-Schwartz Full circle})  we obtain
\begin{equation}\nonumber
	2\pi n^2 \frac{(m^-+h^-r^\alpha)}{ (m^++h^+r^\alpha)^2}\leqslant \oint_{C^r}\left(u_1\frac{\partial u_2}{\partial \vartheta} - u_2 \frac{\partial u_1}{\partial \vartheta} \right)^2 d \lambda
\end{equation}
This lower bound is finite for $r \in [0,R]$ by the assumption from equation (\ref{Holder approximation}), substituting the above equation into the bound from equation (\ref{part 2 interior defect lower bound}) eliminates the dependence on $\vartheta$ yielding
\begin{equation}\label{part 3 interior defect lower bound}
        	\int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})}|u_1\nabla u_2-u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} \geqslant 2\pi n^2 \int\limits_\epsilon^R \left(\frac{m^-+h^-r^\alpha}{m^++h^+r^\alpha}\right)^2\frac{dr}{r}.
        \end{equation}
 We now consider two cases when $m^+ >0$ and when $m^+=0$, in the latter case by our assumption we have that $0< h^- \leqslant h^+$ as otherwise $|\mathbf{x}^1|,|\mathbf{x}^2| = 0$ which implies that $\mathbf{x}$ parametrises a point, not a surface. Additionally if $\alpha =0$ then that implies that $|\mathbf{x}^i|$ is constant in $B_R(\underline{\omega})$ and thus $h^+ = h^- = 0$ and $0<m^-\leqslant m^+$. Consider first case, when $m^+>0$ and $\alpha >0$ then the integral from equation (\ref{part 3 interior defect lower bound}) becomes
        \begin{equation}\nonumber
        	\begin{split}
        		\int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})}|u_1\nabla u_2-u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} \geqslant 2\pi n^2\left(\frac{m^-}{m^+} \right)^2 \log \frac{1}{\epsilon} + 2\pi n^2\left(\frac{m^-}{m^+} \right)^2 \log R  \\ + \frac{2\pi n^2}{\alpha}\left(\left(\frac{h^-}{h^+} \right)^2 - \left(\frac{m^-}{m^+} \right)^2 \right)\log \frac{h^+ R^\alpha + m^+}{h^+\epsilon^\alpha + m^+} \\ + \frac{2\pi n^2 m^+}{\alpha}\left(\frac{h^-}{h^+}- \frac{m^-}{m^+}\right)^2\left(\frac{1}{h^+ R^{\alpha} + m^+} -\frac{1}{h^+ \epsilon^{\alpha} + m^+} \right).
        	\end{split}
        \end{equation}
       Thus as $\epsilon \rightarrow 0$ the Dirichlet integral diverges with rate  $ 2\pi n^2\left(\frac{m^-}{m^+} \right)^2 \log \frac{1}{\epsilon}$ as the lower order terms are constant. Consider the second case, when $m^+>0$ and $\alpha =0$, the lower bound integral becomes
       \begin{equation}\nonumber
        		\int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})}|u_1\nabla u_2-u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} \geqslant 2\pi n^2\left(\frac{m^-}{m^+} \right)^2 \log \frac{1}{\epsilon} + 2\pi n^2\left(\frac{m^-}{m^+} \right)^2 \log R,
        \end{equation}
        which matches the case when $\alpha >0$. Finally, we consider the case when $m^+ = 0$ then we have that equation (\ref{part 2 interior defect lower bound}) becomes
        \begin{equation}\nonumber
        	\int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})}|u_1\nabla u_2-u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} \geqslant 2\pi n^2 \left(\frac{h^-}{h^+}\right)^2\log \frac{1}{\epsilon} + 2\pi n^2 \left(\frac{h^-}{h^+}\right)^2\log R.
        \end{equation}
    \end{proof}
    \subsection{Rate for a boundary defect}
Similar to the previous lemma, we shall construct a lower bound on the energy near a defect, however with the inclusion of the boundary changing the geometry and consequently the bound. Similar to the additional assumptions from section \ref{section: Additional geometery assumptions} we require a higher degree of regularity of the boundary. For a given $\hat{\underline{\omega}} \in \mathbf{x}^{-1}(J_{\mathbf{u}}\cap \partial S)$ we shall assume that there exist $\vartheta_1, \vartheta_2 \in C^1([0,R],\mathbb{R})$ such that $\vartheta_2(t)>\vartheta_1(t)$ for all $t \in [0,R]$ and
\begin{equation}\nonumber
	\mathbf{x}\left( B_r(\hat{\underline{\omega}})\right) \cap \partial S = C_1^r \cup C^r_2, \text{ where } C^r_i:=\left\{\left.\mathbf{x}\left( \hat{\underline{\omega}} + \left(\begin{array}{c} t \cos \vartheta_i(t) \\ t \sin \vartheta_i(t) \end{array} \right)\right)\right| t \in [0,r)\right\}.
\end{equation}
We shall assume that the reciprocal of the difference of the angle functions can be linearised, there exists a constant $\vartheta^{\min} < \vartheta^{\max}$ such that
\begin{equation}\label{linearisation of angle boundary}
	\vartheta^{\min} r \leqslant \frac{1}{\vartheta_2-\vartheta_1}(r) - \frac{1}{\vartheta_2-\vartheta_1}(0) \leqslant \vartheta^{\max}r, \text{ for all }r \in [0,R].
\end{equation}
We understand $\vartheta_2-\vartheta_1(0)$ to be the interior angle of $\Omega$ at the point $\underline{\hat{\omega}}\in \partial \Omega$. We recall that $\mathbf{u}$ is the normalisation of a real analytical vector field, this implies that there exists a constant $u^* \geqslant 0$ such that
\begin{equation}\nonumber
	\left|\int\limits_{C^r_2- C^r_1}u_1 du_2 -u_2du_1 \right| \leqslant u^*r.
\end{equation}
We understand this to be derivative angle between $\mathbf{u}$ and $\mathbf{e}^1$ is bounded, which implies that the boundary data is such that there are no essential discontinuities \cite{EssentialDiscontinuity}. We shall now construct a lower bound on the energy divergence, the majority of the analysis will be similar to the previous lemma.
\begin{lemma}
	\begin{itemize}
\item In the case $m^+\neq 0$ we have that $\lim\limits_{\epsilon \rightarrow 0}\int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})}|u_1\nabla u_2-u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} $ diverges to infinity with rate $\frac{4\pi^2 n^2}{(\vartheta_2-\vartheta_1)(0)} \left(\frac{m^-}{m^+}\right)^2 \log \frac{1}{\epsilon}$.
\item In the case $m^+ = 0$ we have that $\lim\limits_{\epsilon \rightarrow 0}\int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})}|u_1\nabla u_2-u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} $ diverges to infinity with rate $\frac{4\pi^2 n^2}{(\vartheta_2-\vartheta_1)(0)}  \left(\frac{h^-}{h^+}\right)^2 \log \frac{1}{\epsilon}$, where $h^+,h^-$ are the bounds from equation (\ref{Holder approximation}).
	\end{itemize}
\end{lemma}
\begin{proof}
	Let $U_{\epsilon}(\hat{\underline{\omega}}) := \int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})}|u_1\nabla u_2-u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega}$, from the previous proof we can immediately obtain a similar bound to equation (\ref{part 2 interior defect lower bound})
    \begin{equation}\nonumber
    	 U_{\epsilon}(\hat{\underline{\omega}}) \geqslant \int\limits_{\epsilon}^R\left(\int\limits_{\vartheta_1(r)}^{\vartheta_2(r)} \left(u_1\frac{\partial u_2}{\partial \vartheta}-u_2 \frac{\partial u_1}{\partial\vartheta} \right)^2 |\mathbf{x}^{\vartheta}|d\vartheta\right)(m^-+h^-r^{\alpha})\frac{dr}{r}.
    \end{equation}
    Applying Cauchy-Schwartz in a similarly to (\ref{Cauchy-Schwartz Full circle}) we deduce that
    \begin{equation}\nonumber
    	U_{\epsilon}(\hat{\underline{\omega}}) \geqslant \int\limits_{\epsilon}^R\left(\int\limits_{\vartheta_1(r)}^{\vartheta_2(r)} \left(u_1\frac{\partial u_2}{\partial \vartheta}-u_2 \frac{\partial u_1}{\partial\vartheta} \right) |\mathbf{x}^{\vartheta}|d\vartheta\right)^2\frac{1}{(\vartheta_2-\vartheta_1)(r)}\left(\frac{m^-+h^-r^{\alpha}}{m^++h^+r^{\alpha}}\right)^2\frac{dr}{r}.
    \end{equation}
    We recall the definition of the index of a vector field to deduce that
    \begin{equation}\nonumber
    	\left(\int\limits_{\vartheta_1(r)}^{\vartheta_2(r)} \left(u_1\frac{\partial u_2}{\partial \vartheta}-u_2 \frac{\partial u_1}{\partial\vartheta} \right) |\mathbf{x}^{\vartheta}|d\vartheta\right)^2 = \left(2\pi m + \int\limits_{C^1_r-C^2_r}u_1du_2-u_2du_1\right)^2 \geqslant 4\pi^2 n^2 - 4\pi |n| u^* r,
    \end{equation}
    which implies that
    \begin{equation}\nonumber
    	\begin{split}
    	\int\limits_{A^R_{\epsilon}(\hat{\underline{\omega}})}|u_1\nabla u_2-u_2 \nabla u_1|^2|\mathbf{x}^1||\mathbf{x}^2|\ d\underline{\omega} \geqslant \int\limits_{\epsilon}^R\frac{4\pi^2 n^2}{(\vartheta_2-\vartheta_1)(r)}\left(\frac{m^-+h^-r^{\alpha}}{m^++h^+r^{\alpha}}\right)^2\frac{dr}{r} \\ - \int\limits_{\epsilon}^R\frac{4\pi u^* n}{(\vartheta_2-\vartheta_1)(r)}\left(\frac{m^-+h^-r^{\alpha}}{m^++h^+r^{\alpha}}\right)^2dr.
    	\end{split}
    \end{equation}
    Applying the assumption on $\vartheta_2-\vartheta_1$ from equation (\ref{linearisation of angle boundary}) we deduce that
   \begin{equation}\nonumber
   	\begin{split}
   		U_{\epsilon}(\hat{\underline{\omega}}) \geqslant \frac{4\pi^2 n^2}{(\vartheta_2-\vartheta_1)(0)} \int\limits_{\epsilon}^R\left(\frac{m^-+h^-r^{\alpha}}{m^++h^+r^{\alpha}}\right)^2\frac{dr}{r} +4\pi^2 n^2\vartheta^{\min}\int\limits_{\epsilon}^R\left(\frac{m^-+h^-r^{\alpha}}{m^++h^+r^{\alpha}}\right)^2dr \\ - \frac{4\pi u^* |n|}{(\vartheta_2-\vartheta_1)(0)}\int\limits_{\epsilon}^R\left(\frac{m^-+h^-r^{\alpha}}{m^++h^+r^{\alpha}}\right)^2dr- 4\pi u^* |n|\vartheta^{\max}\int\limits_{\epsilon}^R\left(\frac{m^-+h^-r^{\alpha}}{m^++h^+r^{\alpha}}\right)^2rdr.
   	\end{split}
    \end{equation}

We observe that the $2^{\rm{nd}}$, $3^{\rm{rd}}$ and $4^{\rm{th}}$ terms are bounded as $\epsilon \rightarrow 0$ because $m^-\leqslant m^+$ and therefore do not contribute towards the divergence, we may apply the same analysis to the $1^{\rm{st}}$ term as equation (\ref{part 2 interior defect lower bound}), thus obtaining our result.
\end{proof}
\section{Prediction of defect strengths}
Recall that the energy we wish to consider the limit of the energy
\begin{equation}\nonumber
    E_{\epsilon} = \int\limits_{\mathbf{x}(\Omega_\epsilon)}\left|\text{grad}(\mathbf{u})\right|^2 \ d\sigma,
\end{equation}
as $\epsilon \rightarrow 0$. However, as we have just proven if there are any defects in the system then the energy diverges to infinity logarithmically. A reader might think that we need only consider systems that lack defects, but we recall the conservation law for defects given in equation (\ref{Conservation law for defects}), which might force the existence of defects depending on the boundary conditions. Minimising an energy which is infinite would have no meaning, however we can minimise the rate at which the energy diverges. We can predict the strengths of defects within a system using only the geometrical parameters. Note that in liquid crystal systems the strengths of defects can are half-integers \cite{Half_Integer_defect,Defect_Group_theory} and as such we extend our notion of defect strength to include such systems.

The edges of the surface may be designed such that they produce a force on the liquid crystal molecules, which influences the orientation of the molecules. There are two ways to model such an influence, the first is called \textit{Weak Anchoring}, which imposes an energy penalty the further the common axis deviates from the preferred direction. If these forces are sufficiently strong, then we may assume that the vector $\mathbf{u}$ has a fixed direction on the boundary \cite{DeGennes}. One of the most common boundary conditions is conic boundary conditions where the angle between our vector field $\mathbf{u}$ and the tangent to the boundary $\underline{\gamma}'$ (from section \ref{Boundary assumptions}) is constant, with the typical choice being equal to the normal to the edge $\mathbf{u} = \mathbf{N}\times \underline{\gamma}'$ or purely tangential $\mathbf{u} = \underline{\gamma}'$ \cite{Apala_bistable}. Regardless of the specific choice of constant we obtain two important simplifications:
\begin{itemize}
\item Boundary defects are given in terms of the exterior angle at that point \cite{Alex_thesis}, for a point $\underline{s} \in \partial S$ with exterior angle $\tau \in (-\pi,\pi)$ there exists an integer $m \in \frac{1}{2}\mathbb{Z}$ such that $\text{str}(\mathbf{u}, \underline{s}) = m + \frac{\tau}{2\pi}$. Consequently the presence of a vertex implies the existence of a boundary defect.
\item The conservation law from section \ref{Boundary assumptions} is simplified to
\end{itemize}
\begin{equation}\label{Simplified Conservation law}
    \chi = \sum_{k=1}^{\rm{V}_{\partial S}} \frac{\tau_k}{2\pi} + \text{Index}(\mathbf{u},S)
    \end{equation}
As previously established the energy $E_\epsilon$ diverges logarithmically with rate
\begin{equation}\label{Quadratic minimiser full}
	\sum\limits_{\underline{\omega}\in J_{\mathbf{u}}\cap \Omega}  q(\underline{\omega}) n^2(\underline{\omega}) + \sum\limits_{\hat{\underline{\omega}}\in J_{\mathbf{u}}\cap \partial \Omega} \frac{2\pi}{\pi - \phi(\underline{\omega})} q(\underline{\omega}) n^2(\underline{\omega}),
\end{equation}
where the function $q: \overline{\Omega} \rightarrow [0,\infty)$ is given by
\begin{equation}\nonumber
	q(\underline{\omega}) = \left\{ \begin{array}{cl}
    \frac{ \min\{|\mathbf{x}^1|,|\mathbf{x}^2|\} }{\max\{|\mathbf{x}^1|,|\mathbf{x}^2|\} }(\underline{\omega}), & \max\{|\mathbf{x}^1|,|\mathbf{x}^2|\} (\underline{\omega})\neq 0,\\
    \frac{ h^-}{h^+}, &\max\{|\mathbf{x}^1|,|\mathbf{x}^2|\} (\underline{\omega}) =  0,\\
	\end{array}\right.
\end{equation}
for $h^- < h^+$ are given in equation (\ref{Holder approximation}). The function $n:\bar{\Omega}\rightarrow \mathbb{R}$ defined to be the pointwise limit of the index,
\begin{equation}\nonumber
	n(\underline{\omega}) := \lim\limits_{r\rightarrow 0} \text{Index}(\mathbf{u}, \mathbf{x}(B_r(\underline{\omega})\cap \Omega)).
\end{equation}
The function $\phi: \partial \Omega \rightarrow (-\pi,\pi)$ is the exterior angle of $\partial \Omega$, not to be confused with $\tau$ which is the exterior angle of $\partial S$. Thus to minimise the rate at which the energy diverges to infinity one needs to minimise (\ref{Quadratic minimiser full}) with respect to (\ref{Simplified Conservation law}). Recall that the boundary defects can be expressed as a constant plus an integer which implies that the domain of both our function and constraint are the half-integers.

We can consider a simplification of the above system when $S$ is a planar domain and thus $|\mathbf{x}^1|=|\mathbf{x}^2| = 1$ and the exterior angle of $\partial S$ is equal to $\partial \Omega$, then our quadratic minimisation problem becomes the minimisation of $Q: \frac{1}{2}\mathbb{Z}^{ |J_{\mathbf{u}}|}\rightarrow [0,\infty)$
\begin{equation}\nonumber
	Q = \sum\limits_{i=1}^{|J_{\mathbf{u}}\cap \Omega|}n_i^2 + \sum\limits_{j=1}^{\rm{V}_{\partial S}} \frac{2\pi }{\pi - \tau_j}\left(m_j - \frac{\tau_j}{2\pi} \right)^2, \text{ subject to } \chi = \sum\limits_{i=1}^{|J_{\mathbf{u}}\cap \Omega|}n_i + \sum\limits_{j=1}^{\rm{V}_{\partial S}} m_j.
\end{equation}
A great example of how this principle can be applied is when we consider the square well with nematic liquid crystals \cite{Apala_bistable}, such a system has two steady states the rotational and diagonal state, with no interior defects and defects of $\{\frac{1}{2}, \frac{1}{2}, 0, 0\}$ at each of the vertices. The locations of the defects in the vertices determine what "state" the system is in, but the question of why there are no other families of solutions can be explained through the above minimisation principle.

Similarly we can predict the families of solutions for any other reasonably smooth system, for example consider an equilateral triangle, which implies that $N=3$, $\chi =1$ and $\tau_i = \frac{2\pi}{3}$ for $i=1,\ldots,3$. Using a numerical minimiser we deduce that "minimising" configuration has three defects of strength $+\frac{1}{6}$ located at the vertices and one internal defect of strength $-\frac{1}{2}$, which is illustrated in figure (\ref{fig: Triangle}).
\begin{figure}
	\centering
    \includegraphics[width = 0.4\textwidth]{Triangle.png}
    \caption{An illustration of the optimal defect configuration for a planar triangle.}
    \label{fig: Triangle}
\end{figure}
This piece has been focused on the Oseen-Frank model on the surface of a manifold. However, similar work has been conducted on the Landau-de Gennes model in planar polygonal domains. We can compare the configuration of defects predicted by the derived algorithm against the results that these authors find.

In the case of a equilateral triangular well, the configuration of defects matches precisely the configurations found by Han, Majumdar and Zhang \cite{HanYucen2020ARSf} for both the large and small $\lambda$ parameter. In the context of two dimensional modelling the large $\lambda$ limit of the Landau-de Gennes model corresponds to the Oseen-Frank one constant approximation.

In the case of an equilateral hexagonal well, the work of Han, Harris, Majumdar and Walton predict the existence of multiple branches of defect patterns.
\begin{itemize}
\item The three branches denoted \textit{Ortho}, \textit{Meta} and \textit{Para}, correspond to permutations of the corner defect strengths: two of strength $\frac{1}{3}$ and four of strength $-\frac{1}{6}$ \cite{HanYucen2021TNaM, HanYucen2020ARSf}, these branches are present in the limit $\lambda\rightarrow \infty$.
\item The $M1$-states which correspond to a single internal defect of strength $\frac{1}{2}$, one corner defect of strength $\frac{1}{3}$ and five corner defects of strength $-\frac{1}{6}$ \cite{HanYucen2020Sloa}.
\item The \textit{BD}-state which corresponds to corresponds to six corner defects of strength $-\frac{1}{6}$ and two internal defects of strength $\frac{1}{2}$.
\end{itemize}
These five branches are all predicted by the minimisation method we have described as they all possess the same energy divergence rate. However, these are not the only branches of defect patterns which are observed in the Landau-de Gennes model.

The \textit{Ring} pattern, which is similar to the \textit{BD}-state because of the six corner defects of strength $-\frac{1}{6}$, has a single internal defect of strength $+1$. This defect pattern is not predicted, as the minimisation principle considers the sum of the squares of the defect strengths, and the internal defect of +1 has a larger rate of energy divergence than two defects of strength $\frac{1}{2}$.

The transitional states which are denoted  $T$ and $H$, similarly are not observed as, the perspective of the defect configuration identical to one of the other states but with additional pairs of internal defects of strengths $-\frac{1}{2}$ and $\frac{1}{2}$. Each pair of defects, while possessing a net defect strength of $0$, increase the rate of divergence by $+\frac{1}{2}$. Thus this minimisation principle prefers the \textit{BD}-state to both the $T$ and $H$ states.

Finally, we consider a classical non-planar domain: the sphere. The absence of a boundary implies that there are only internal defects and that the sum of their strengths is equal to two, the Euler characteristic of a sphere. The minimisation principle implies we should observe four defects each of strength $+\frac{1}{2}$. This forms a \textit{baseball} like pattern, where the defects are placed on the surface equidistant from each other, as if they were the vertices of a tetrahedron. This particular state has been theorised and numerically investigated before \cite{NematicShellSphere1,NematicShellSphere2}, which lends credit to the minimisation principle. However, there is an additional state, which has been physically observed \cite{Boundary_defects_Boojums}, consisting of a pair of $+1$ defects one acting as the \textit{source} and the other as the \textit{sink} for the vector field. This is not predicted by the algorithm, since two defects of strength $+1$ has a higher rate of divergence than four defects of strength $+\frac{1}{2}$.

\section{Conclusion and Discussion}
We derived a conservation law for vector fields on the surface of a manifold using the notions of triangulation and an application of the Gauss-Bonnet theorem. We then considered the Oseen-Frank one constant approximation (the Dirichlet integral) on the surface of the manifold and investigated the rate at which the energy density diverges locally because of the defect. Finally, we combined the two results to create a discrete minimisation problem with a linear constraint, the solutions of which predict the strengths of defect branches. Finally, we compared the predictions that this theorem made with results from practical experiments and from numerical simulations.

In future work, the derivation of the generalisation of the Poincaré–Hopf theorem that we have presented could potentially be generalised further for surfaces which can be described by an atlas of charts, rather than a single one. The proof sketch would involve applying the previous theorem on each chart and then simplifying the sum of angles in a similar method to equation (\ref{appendix: angle sum intermediate}).

The prediction theorem we have presented is valid for the Oseen-Frank one constant approximation, which (for thermotropic liquid crystals) is the limit of the Landau-de Gennes model as the temperature decreases \cite{Brezis}. However, we have seen that this heuristic prediction theorem cannot capture certain states. At the present time it is unclear whether this is because the prediction theorem is lacking an important term or if the states are unstable in the low temperature limit.

In this piece we considered the leading order term of the lower bound of the localised Dirichlet integral about each defect, which allowed us to derive a heuristic prediction about the defect strengths. This method predicts the various branches of defects, however it does not give any information about which branch is "preferred". For example, in the shallow rectangular well the diagonal state may energetically preferable to the rotated state \cite{Apala_bistable,Alex_thesis} depending on the ratio between the lengths of the sides. A potential continuation would be to consider the lower order terms of this energy, this may allow us to heuristically predict the locations of the internal defects and understand which branches are energetically preferable.

Despite the issues we have discussed, the ability to predict potential defect configurations based solely on geometrical parameters could have extensive uses in numerical analysis to construct well posed initial conditions, which are essential in finite element methods. Additionally, the predictive method can be applied to unusual, non-regular, non-polygonal well shapes making it exceptionally versatile. Finally, the algorithm to predict the defect strengths is numerically easy to apply; as it is the discrete minimisation of a quadratic with respect to a linear constraint; where the coefficients are geometrical constants. Such an algorithm is so numerically cheap, the calculations may be done on a smartphone within a few seconds.

\subsection{Acknowledgements}
We are grateful for the financial support of the University of Bath's department of Mathematical Sciences. Additionally, we are thankful for the support and guidance of Kirill Cherednichenko, Apala Majumdar and Jey Sivaloganathan.
\printbibliography
\end{document}

redpajama_set_name: RedPajamaArXiv


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I don't think Robert is involved in that scandal.
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Rayan grudgingly did what Carisa asked.
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Maybe Spy's got something to hide.
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Ethan sat on the bench and crossed his legs.


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text:
Home / Security / News
FBI says hack tool only works on iPhone 5c
FBI Director James Comey said the FBI bought the tool.
By John Ribeiro
PCWorld	 Apr 7, 2016 3:03 am PDT
Only the iPhone 5c running iOS 9 can be unlocked by the tool the FBI bought to crack the iPhone used by one of the San Bernardino killers.
The tool does not work on the iPhone 5s or 6, so it only addresses a "narrow slice" of iPhones, Federal Bureau of Investigation Director James Comey said late Wednesday at Kenyon College.
The government is considering whether it should disclose to Apple the flaw that aided the hack: "We just haven't decided yet," he said at the Ohio college's Center for the Study of American Democracy.
A court in California ordered Apple to help the FBI to hack by brute force the passcode of the iPhone 5c. The government was concerned that, if an auto-erase feature was activated on the phone, the data that the FBI was looking for would be automatically erased after 10 unsuccessful attempts, so it wanted a workaround from Apple.
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The agency has to date not disclosed the name of the organization or the hack that was used.
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Comey said the FBI is very good at keeping secrets and the agency has a high amount of confidence that the people it bought the tool from are very good at protecting it.
In another lawsuit in New York, the FBI is demanding that Apple should help it extract data from a locked iPhone 5s running iOS 7 used by an alleged drug dealer. The government has appealed the case after a Magistrate Judge ruled in Apple's favor.


text:
The Girl Can't Help It is a 1956 American musical comedy film starring Jayne Mansfield in the titular role, Tom Ewell, Edmond O'Brien, Henry Jones, and Julie London.  The picture was produced and directed by Frank Tashlin, with a screenplay adapted by Tashlin and Herbert Baker from an uncredited 1955 short story, "Do Re Mi" by Garson Kanin. Filmed in DeLuxe Color, the production was originally intended as a vehicle for the American sex symbol Jayne Mansfield, with a satirical subplot involving teenagers and rock 'n' roll music. The unintended result has been called the "most potent" celebration of rock music ever captured on film.

The original music score, including the title song performed by Little Richard, was by Bobby Troup, with an additional credit to Ray Anthony for the tune "Big Band Boogie".

Plot
A slot machine mobster, Marty "Fats" Murdock, wants his blonde girlfriend, Jerri Jordan, to be a singing star, despite her seeming lack of talent. He hires alcoholic press agent Tom Miller to promote Jordan, both because of his past success with the career of singer Julie London and because he never makes sexual advances towards his female clients.

Miller sets to work by showing Jordan off around numerous night spots; his machinations arouse interest in Jordan and soon offers of contracts follow. However, Miller realizes that Jordan really just wants to be a homemaker and tries to persuade Murdock not to push Jordan into a show-business career. He thinks he's succeeded when he reveals to Murdock that Jordan's singing is so bad it shatters light bulbs, but Murdock suggests that Jordan would be perfect for a song he composed while in prison, "Rock Around the Rock Pile" (a parody of "Rock Around the Clock"). Miller reluctantly records Jordan performing the part of a prison siren in Murdock's song and heads to Chicago to promote it to Wheeler, a former mob rival of Murdock who now has a monopoly over the jukebox industry.

Suspicious of Miller's reluctance to promote Jordan and of the obvious attraction between Miller and Jordan, Murdock has his associate Mousie wiretap a phone call between the pair. Feeling pity for them, Mousie edits out the romantic portions of their conversations and convinces Murdock that their relationship is strictly business.

In Chicago, Wheeler is impressed by the song and Jordan's voice and offers to sign both Jordan and the songwriter. However, when Miller reveals that the songwriter is Murdock, Wheeler throws him out of his office and vows never to play the song. A furious Murdock bullies bar owners into buying jukeboxes from him instead and successfully promotes his and Jordan's song. To prevent Murdock from stealing his business, Wheeler arranges to have Murdock assassinated at the rock show where Jordan will be making her debut.

On his way to the show, Murdock confesses to Mousie that he doesn't want to marry Jordan. Mousie confesses that he altered the tape of Jordan and Miller's phone call and encourages Murdock to let Jordan marry Miller. Backstage at the show, Jordan confesses her love to Miller and they kiss. Jordan also admits that she is a talented singer, who lied because she did not want a show business career; she goes on stage and performs a song about her love for Miller. When Murdock arrives, Miller declares to him that he and Jordan are in love; the delighted Murdock surprises Miller by shaking his hand and offering to be the best man.

Before Miller and Murdock can tell Jordan the good news, Wheeler's assassins shoot at Murdock. Miller fights them off and shoves Murdock on stage to perform his song, reasoning that the assassins won't shoot Murdock in front of so many witnesses. Wheeler arrives and, impressed by the audience's response to Murdock, calls off the assassination and signs Murdock instead. The film ends with Miller and Jordan kissing on their honeymoon, as Murdock and Mousie perform on a TV show in the background.

Cast

Production
"Do Re Mi", a short story by Garson Kanin that had appeared in The Atlantic magazine in March 1955, was acquired by Fox. They initially assigned producer Nunnally Johnson to develop the film, but then decided to emphasize rock 'n' roll in it and, in 1956, reassigned it to director Frank Tashlin, whose background included work with music in animation. Tashlin collaborated on a new script with Herbert Baker that played up visual humor that was virtually cartoonish. Kanin did not approve of the new take on his story and requested his name be removed from the credits. Subsequently, Tashlin came up with the new title, The Girl Can't Help It.

Reception
The film received mixed reviews by critics. Bosley Crowther of The New York Times noted that Mansfield's performance was underwhelming, stating, "Her range, at this stage, appears restricted to a weak imitation of Marilyn Monroe."

Accolades
The film is recognized by American Film Institute in these lists:
 2006: AFI's Greatest Movie Musicals – Nominated

Influence on rock music
The film's influence on rock music is significant. The film reached Liverpool, England, in the early summer of 1957. The cameo performances of early rock 'n' roll stars such as Little Richard, Eddie Cochran, and Gene Vincent and His Bluecaps fascinated a 16-year-old John Lennon by showing him, for the first time, his "worshipped" American rock 'n' roll stars as living humans and thus further inspiring him to pursue his own rock 'n' roll dream. On July 6, 1957, 15-year-old Paul McCartney was introduced to Lennon after the latter had performed at a village church garden party with his skiffle group The Quarrymen. McCartney demonstrated his musical prowess to Lennon by performing "Twenty Flight Rock" in a manner similar to how he had seen it played by Eddie Cochran in The Girl Can't Help It. This led to Lennon inviting McCartney to join the group. McCartney talks about the film in the documentary series The Beatles Anthology.

On September 18, 1968, The Beatles interrupted recording "Birthday" at Abbey Road Studios so they could go back to Paul McCartney's house to watch the British TV premiere of the film.

Also, some film buffs have pointed to Elvis Presley's famous performance of the song "Jailhouse Rock" in MGM's film of the same name (often cited as the first music video), released one year after The Girl Can't Help It, as bearing a remarkable resemblance to the theme and performance of a song called "Rock Around the Rockpile" from the earlier film. In that performance, Edmond O'Brien plays a character who seeks to escape an assassination attempt by jumping on stage and singing the lyrics, "rock, rock, rock around the rockpile," while backed up by The Ray Anthony Band wearing striped inmate uniforms. O'Brien, then 42 years old, even awkwardly attempts some of the hip-swiveling and leg motions for which Elvis had already become famous, as he had appeared already 10 times, to cumulative audiences in excess of 180 million viewers, and on national television, before and during the filming of The Girl Can't Help It, the production of which commenced in mid-September 1956. Other film buffs point to the famous acrobatic dancer and Presley contemporary, as well as MGM star, Russ Tamblyn (Seven Brides for Seven Brothers), who visited Presley at his suite at the Knickerboker Hotel on the night before the filming of the "Jailhouse Rock" scene. Presley had had trouble that day at the rehearsals in interpreting what the MGM choreographer, Alex Romero, demanded of him, and Tamblyn's advice at the suite that night helped him perform the first part of that dance sequence (which, incidentally, was witnessed by the then-MGM megastar Gene Kelly).  In fact, The Girl Can't Help It'''s producers had sought to enlist Presley, whose manager, Colonel Tom Parker, however, had demanded too much money. Two uncredited composers on The Girl Can't Help It, Hugo Friedhofer and Lionel Newman, had also composed music for the Elvis film Love Me Tender'', in the same year, 1956.

Songs performed in the film
 "The Girl Can't Help It" – Little Richard
 "Tempo's Tempo" – Nino Tempo
 "My Idea of Love" – Johnny Olenn
 "I Ain't Gonna Cry No More" – Johnny Olenn
 "Ready Teddy" – Little Richard
 "She's Got It" – Little Richard
 "Cool It Baby" – Eddie Fontaine
 "Cinnamon Sinner" – Teddy Randazzo and the Three Chuckles
 "Spread the Word" – Abbey Lincoln
 "Cry Me a River" – Julie London
 "Be-Bop-A-Lula" – Gene Vincent and His Blue Caps
 "Twenty Flight Rock" – Eddie Cochran
 "Rock Around the Rockpile" – Edmond O'Brien; Ray Anthony and his Orchestra
 "Rockin' Is Our Business" – The Treniers
 "Big Band Boogie" – Ray Anthony and his Orchestra
 "Blue Monday" – Fats Domino
 "You'll Never, Never Know" – The Platters
 "Ev'ry Time (It Happens)" – Jayne Mansfield (dubbed by Eileen Wilson)
 "Giddy Up a Ding Dong" – Freddy Bell & The Bell-Boys

References

Bibliography

External links

1956 films
1956 musical comedy films
1950s crime comedy films
1950s satirical films
20th Century Fox films
American crime comedy films
American musical comedy films
American rock musicals
American satirical films
1950s English-language films
Films about singers
Films based on American novels
Films directed by Frank Tashlin
Films with screenplays by Frank Tashlin
CinemaScope films
1950s American films

redpajama_set_name: RedPajamaWikipedia


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Home » Magazine » Bulgari Diagono Scuba 2016 Watch
Bulgari Diagono Scuba 2016 Watch
Written by Bojan Pavlovic on August 16, 2016 in Magazine, News & Events - No comments
The original Diagono Scuba model was introduced by Bulgari 22 years ago. Since that time, this diver's watch has gained extreme popularity, particularly in Italy during 1990s. Last year, the iconic piece was reworked and now the line has been enriched with two blue variants.
Bulgari Diagono Scuba 2016 is designed to be equally appropriated for various formal and informal occasions. Like its predecessors, this newcomer features large glowing indexes that have even been increased if compared to the original model. This is quite improvement considering the fact that the glowing degree and the visibility degree are two fundamental features of the scuba-diving watch.
The new Diagono Scuba is housed in a round stainless steel case which measures 41mm in diameter and incorporates the unidirectional ratcheted bezel which serves to measure dive and decompression times, as well as the winding crown which has been simplified by inserting a different special groove that has also been embellished in the middle by a round ceramic insert. The exceptional diver boasts an exceptional level of water resistance – up to 300 meters (approximately 1,000 feet).
Underneath the front sapphire glass lies the ocean blue dial enhanced with the Superluminova-treated hour markers and large Arabic numerals 6 and 12. A date window is situated at the 3 o'clock position. Unexpectedly, the black date disk goes perfectly with the blue dial.
Bulgari Diagono Scuba is powered with the caliber BVL 191 self-winding movement that operates at the rapid frequency of 28,800 alternations per hour (4 Hz) and can store enough power to keep the watch ticking for about 42 hours when fully wound. The mechanism is tooled with a stop-seconds device which serves to block the balance when the crown is pulled out and with a disconnecting system that avoids wear of the pinions and saves power when the winding device is activated, either via the winding crown or by the motion of the oscillating weight which is mounted on ceramic ball bearings.
The impeccable caliber is in-house manufactured at the Bulgari's facility in La Chaux-de-Fonds, Switzerland. The movement's finishes also include Côtes de Genève decoration.
Depending on the model, the new Bulgari Diagono Scuba comes with either a blue rubber strap with a steel pin buckle or a steel bracelet, fitted with a steel folding clasp. The first model (ref. 102504 DP41C3SVSD) can be purchased for Euro 6,050 (approximately $6,550), while the latter (ref. 102586 DP41C3SSSD) is priced at Euro 6,550 ($7,100).


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