text:
\section{Introduction and main results}

\indent
Since Schramm introduced Schramm-Loewner evolution (SLE) in a simply connected domain \cite{MR1776084}, SLE theory has successfully produced rigorous proofs of several remarkable conjectures in statistical physics over the past two decades.
Among these, conformal invariance of scaling limits of some important critical lattice models has been proved by means of SLE.
Such results have been obtained by the fundamental fact that certain discrete observables converge to conformally covariant martingale processes. For instance, in \cite{MR1851632}, Smirnov used Cardy's observables to prove conformal invariance of scaling limit of critical site percolation in a hexagonal lattice.
Schramm and Sheffield made use of a certain bosonic observable to prove that the level lines of discrete Gaussian free field (GFF) with specific height gaps converges to SLE(4), see \cite{MR2486487}. Chelkak, Duminil-Copin, Hongler, Kemppaninen and Smirnov proved existence of scaling limit of FK Ising model interfaces and its conformal invariance by virtue of certain parafermionic observables, see \cite{MR3151886} and references therein.
In addition, some geometric properties of SLE curves have been obtained by means of SLE martingale-observables. Examples include the study of Hausdorff dimensions (Beffara's observables), and left passage probabilities (Schramm's observables) of SLE, see \cite{MR2435854, MR1871700}.

In this paper we use the framework developed in \cite{MR3052311,kang2012radial,MR3123602,MR3073040,KM17} to implement a version of conformal field theory (CFT) in a doubly connected domain with excursion reflected (ER)/Dirichlet boundary conditions and construct a large class of martingale-observables for variants of annulus $\SLE.$
In the physics literature, it is well known that under the insertion of ratio of the one-leg operator $\Psi$ to its correlation function $\E\,\Psi$, the correlators in a certain class of fields are martingale-observables for SLEs.
In a doubly connected domain, Hagendorf, Bernard and Bauer studied GFF with Dirichlet/Dirichlet-Neumann boundary conditions, and described some martingale-observables for specific types of annulus SLE(4), see \cite{MR2651436}.
Hagendorf presented physical arguments on how to extend this result to $\kappa\ne4$ for the Dirichlet boundary case in \cite{1742-5468-2009-02-P02033}.

In mathematics, Zhan has studied basic properties of SLE in a doubly connected domain. For $\kappa \le 4,$ he considered the annulus $\SLE(\kappa,\Lambda)$ with one force point, where $\Lambda := \kappa (\log Z)'$ is the drift function of driving process of SLE and introduced the null vector equation (parabolic PDE) for the partition function $Z$ such that $\SLE(\kappa,\Lambda)$ satisfies domain Markov property and reversibility, see \cite{MR3334276}. Moreover, in \cite{MR2902453}, he showed that these SLEs satisfy the generalized restriction property. At the same time, Lawler used the generalized restriction property and Brownian loop measure to define SLEs in a multiply connected domain and showed that in the annulus, his crossing SLE partition function (the total mass of a positive measure on crossing SLE curves) for $\kappa\le 4$ satisfies the null vector equation, see \cite{lawler2011defining}.

The statistical fields we consider in this paper form the OPE family $\mathcal{F}_{(b)}$ generated by central charge modifications $\Phi_{(b)}$ of the GFF in a doubly connected domain under OPE multiplication. Here, $b$ is a real parameter related to the central charge $c$ as $c=1-12b^2$ and to the SLE parameter $\kappa$ as
$b = \sqrt{\kappa/8}-\sqrt{2/\kappa}.$
We consider two types of boundary conditions of GFF, ER and Dirichlet boundary conditions.
Green's function with ER boundary condition or Green's function for excursion reflected Brownian motion (see \cite{MR2992562}) is known as a \emph{modified Green's function} in the analysis literature, e.g., see \cite{MR2241034}.

Ward's equation is one of our main ingredients to establish the relation between CFT in a doubly connected domain and annulus SLE. Whereas Ward's equations in a simply connected domain describe the Virasoro fields in terms of Lie derivative operators within correlations, the statement of those in a doubly connected domain contains the derivative of correlation functions with respect to the modular parameter as well. These forms of Ward's equations are well known in the physics literature, e.g., Ward's equations in a doubly connected domain \cite{MR2651436} and Eguchi-Ooguri's version of Ward's equations on a complex torus of genus one \cite{MR873031}.
We derive them not from the path integral formalism but from the pseudo-addition theorem of Weierstrass zeta function.
See \cite{KM17} for a mathematical derivation of Eguchi-Ooguri's version of Ward's equations.

We construct the one-leg operator $\Psi_{\bfs\beta}\equiv\Psi_{\bfs\beta}(p,\bfs q)$ from the Gaussian free field as the modified multi-vertex field (or the OPE exponential) with the charge $a=\sqrt{2/\kappa}$ at the starting point $p$ of $\SLE(\kappa,\Lambda)$ and additional charges $\bfs\beta = \sum \beta_j\cdot q_j$ at force points $\bfs q.$ Here we require that total sum of charges should vanish. On a complex torus (of genus one), this requirement is known as the neutrality condition in the physics literature. With this choice of charge at $p,$ $\Psi_{\bfs\beta}$ satisfies the level two degeneracy equation. Following the set-up in the physics literature, we set $\Lambda(\xi,\bfs q) = \kappa \,\pa_\xi \log \E\,\Psi_{\bfs\beta}(\xi,\bfs q)$ and consider a collection of all correlation functions of fields in the OPE family $\FF_{(b)}$ generated by $\Phi_{(b)}$ under the insertion of the properly normalized one-leg operator. Then we show that this collection forms martingale-observables for $\SLE(\kappa,\Lambda),$ see Theorem~\ref{main} below.
For Dirichlet boundary condition, the neutrality condition for $\Psi_{\bfs\beta}(p,\bfs q)$ coincides with the boundary condition that Izyurov and Kyt\"{o}l\"{a} considered in \cite{MR3010393}, i.e., the requirement that jumps (of the correlation function $\varphi(z):=\E\, \Phi_{(0)}(z) \Psi_{\bfs\beta}(p,\bfs q)/\E\, \Psi_{\bfs\beta}(p,\bfs q)$) on the boundary should add up to zero. They explained that the 1-point function $\varphi$ and the 2-point function $(z_1,z_2)\mapsto\E\, \Phi_{(0)}(z_1)\Phi_{(0)}(z_2) \Psi_{\bfs\beta}(p,\bfs q)/\E\, \Psi_{\bfs\beta}(p,\bfs q)$ are martingale-observables for $\SLE(4,\Lambda).$
However, the null vector equation for the partition function $Z_{\bfs\beta}:=\E\,\Psi_{\bfs\beta}$ of this type of $\SLE(\kappa,\Lambda)$ with one force point is different from Lawler and Zhan's unless $\kappa=4.$

We apply the method of screening to find an explicit solution to Lawler and Zhan's null vector equation for each $\kappa$ in the chordal case and to construct a class of $\SLE(\kappa,\Lambda)$ martingale-observables associated with this solution. If $\kappa > 4,$ we can deform a Pochhammer contour in the method of screening so that  the solution can be represented in terms of Euler type integral. In the case $\kappa \le 4,$ we perform a proper analysis on an analytic continuation (with respect to $\kappa$) of the Pochhammer contour integral to obtain a non-trivial real-valued solution. We remark that in \cite{MR3334276}, Zhan described a solution in the crossing case with $\kappa \le 4$ in terms of the Feynman-Kac formula.

\begingroup
\setcounter{tmp}{\value{thm}}
\setcounter{thm}{0}
\renewcommand\thethm{\Alph{thm}}

\subsection{Notation}

Throughout this paper, we use the following symbols: let $\T= \{ z \in \C : |z|=1 \}$; for $r>0$ let $\mathbb A_r = \{z \in \C: e^{-r} < |z| <1 \},$ $\T_r= \{ z \in \C : |z|=e^{-r} \},$ $\S_r = \{z \in \C: 0 <\Im\,z < r \},$ and $\R_r= \{ z\in \C: \Im\,z=r \}.$ We also denote by $\mathcal C_r$ the cylinder $\S_r / \langle z \mapsto z+2 \pi \rangle,$ and by $D_r$ a doubly connected domain with modulus $r.$
A subset $K$ of a doubly connected domain $D$ is called a hull in $D$ if $D\sm K$
is a doubly connected domain and $K$ has positive distance from one boundary component
of $D.$ For any hull $K$ of $D,$ let us define $\text{cap}_D(K) := \text{mod}(D)- \text{mod} (D \sm K)$ as the capacity of $K$ in $D,$ where $\text{mod}(\cdot)$ denotes the modulus of a doubly connected domain. We write $\Theta(r,z):=\theta \left(\frac{z}{2\pi},\frac{ir}{\pi} \right), \ti{\Theta}(r,z):=\Theta(r,z)\,e^{z^2/4r} $ and $\Theta_I(r,z):=\theta_2 \left(\frac{z}{2\pi},\frac{ir}{\pi} \right)$, where $\theta(z,r)$ and $\theta_k(z,r),$ $(k =1, 2, 3)$ are Jacobi theta functions, see \cite[p.63]{MR808396}. For a notational convenience, we sometimes omit the modular parameter $r$, e.g. $\Theta(\cdot)\equiv\Theta(r,\cdot)$, etc.

\subsection{Basic setup}

For a fixed real parameter $b$, we define the \textit{central charge modification} of GFF $\Phi_{(0)}$ in $D_r$ by adding a certain non-random pre-pre-Schwarzian form as follows:
\begin{equation*}
\Phi_{(b)}:=\Phi_{(0)}-2b\arg w',
\end{equation*}
where $w$ is a conformal map from $D_r$ onto $\mathcal C_r.$
We remark that $\arg w^{\prime}$ does not depend on the choice of a conformal map.
We denote by $\FF_{(b)}$ the OPE family of the $\Phi_{(b)},$ the algebra (over $\C$) spanned by the generators $1,$ mixed derivatives of $\Phi_{(b)},$ those of OPE exponentials $e^{\ast \alpha \Phi_{(b)}}$ ($\alpha \in \C$), and their holomorphic/anti-holomorphic/mixed parts (with certain neutrality conditions) under the OPE multiplication.

We now introduce the annulus SLE with $N$-marked boundary points $q_1,\cdots q_N.$
Similarly as in a simply connected domain, the annulus $\SLE(\kappa,\Lambda)$ can be described by Loewner's differential equation, which was introduced by Zhan in \cite{zhan2004stochastic}.
Let $\xi$ be a (real-valued) continuous function defined on $[0,T)$ for some $0< T <r$.
By definition, the Loewner kernel $S(r,z)$ on $\A_r=\{z: e^{-r}<|z|<1\}$ is given by
$$
S(r,z)=\lim_{N \rightarrow \infty}\sum_{n=-N}^{N} \frac{e^{2nr}+z}{e^{2nr}-z}.
$$
 For each $z \in \A_r$, let $g_t(z)$ be the solution (which exists up to the first time $\tau_z\in(0,T]$ that $g_t(z)$ hits $\xi_t$) of the annulus Loewner equation
\begin{equation*}
\displaystyle	\pa_t g_t(z) =g_t(z) S(r-t, g_t(z)/e^{i\xi_t}), \quad g_0(z)=z.
\end{equation*}
For each time $t \in [0,T)$, let $K_t := \{ z \in \A_r : \tau_z\leq t\}.$ Then $K_t$ is a hull in $\A_r$ and has capacity $t$. Moreover, $g_t$ is a conformal map from $(\A_r \sm K_t,\T_r)$ onto $(\A_{r-t},\T_{r-t})$.

It is convenient to describe annulus SLE in the covering space $\S_r$, where the Loewner kernel $H$ is expressed in terms of $\Theta$ as
$$
H(r,z):=-i \, S(r,e^{iz})=2 \, \pa_z \log \Theta(r,z).
$$
For each $z\in \S_r$, we denote by $\ti{g}_t(z)$ the solution of the equation
 \begin{align*}
 \pa_t \ti{g}_t (z)=H(r-t,\ti{g}_t(z)-\xi_t), \quad \ti{g}_0(z)=z,
 \end{align*}
and set $\ti{K}_t := \{ z \in \S_r : \tau_z\leq t\}$. Then $\ti{g}_t$ is a conformal map from $(\S_r \sm \ti{K}_t; \R_r)$ onto $(\S_{r-t};\R_{r-t})$. Moreover for each $z \in \S_r \sm \ti{K}_t$,
$e^{i\ti{g}_t(z)}=g_t(e^{iz})$, and $K_t=\{   e^{iz}\in \A_r : z\in \ti{K}_t   \}. $
We call that $\xi$ generates a trace $\gamma$ if
 $$
 \gamma_t=\gamma(t):=\lim_{z \rightarrow \xi_t} \ti{g}_t^{-1}(z).
 $$
It was proved by Zhan in \cite{zhan2004stochastic} that $\xi_t:=\sqrt{\kappa}B_t$ ($t<r$) a.s. generates continuous trace, where $B_t$ is a standard one-dimensional Brownian motion.

Suppose $\Lambda$ is a real-valued $C^{0,1}$ function on $(0,\infty)\times (\C \sm \{ z \in 2\pi\mathbb{Z} +2ir\mathbb{Z} \})^N$ with period $2\pi$ in each space variable. Let $\xi_t$ be the solution, which exists up to the first time $T$ such that $\xi_t-\tilde{g}_t(q_j) \in 2\pi \mathbb{Z}$, of the SDE
 \begin{equation*}
 d\xi_t =\sqrt{\kappa}dB_t +\Lambda \left(r-t,\xi_t-\ti{g}_t(q_1),\cdots,\xi_t- \ti{g}_t(q_N) \right)	dt, \quad \xi_0=p.
 \end{equation*}
Applying the Girsanov theorem, it can be shown that up to time $T$, $\xi_t$ a.s. generates continuous trace, see \cite[Section 3.2]{MR3334276} for further details.
We call the trace generated by $\xi_t$ the annulus $\SLE(\kappa,\Lambda)$ trace started from $p$ with force points $q_1.\cdots,q_N.$
\begin{comment}
\begin{def*}
	Suppose $\Lambda$ is an annulus drift function. For $\kappa >0,$ $r>0,$ $p \in \R,$ and $q_1,\cdots q_N \in \pa \S_r,$ let $\xi_t$ ($0\leq t <r$) be the maximal solution of the SDE
	\begin{equation*}
	d\xi_t =\sqrt{\kappa}dB_t +\Lambda \left(r-t,\xi_t-\ti{g}_t(q_1),\cdots,\xi_t- \ti{g}_t(q_N) \right)	dt,\quad \xi_0=p,
	\end{equation*}
	where $B_t$ is a standard one-dimensional Brownian motion with $B_0=p$ and $\ti{g}_t$ are covering annulus Loewner maps driven by $\xi$ in $\S_r.$ Then the trace generated by $\xi$ is called the \textit{covering annulus $\SLE(\kappa,\Lambda)$ trace} in $\S_r$ started from $p$ with the marked boundary points $q_1.\cdots,q_N.$
\end{def*}
\end{comment}

Now we recall the definition of SLE martingale-observables.
 A non-random (conformal) field $f$ in a planar domain $D$ (or on a Riemann surface $M$ in general) is an assignment of smooth function
$ \left( f \|\phi \right) : \phi U \rightarrow \C$
for each local chart $\phi.$
By definition, a non-random field $M$ is a \textit{martingale-observable} for annulus $\text{SLE}(\kappa,\Lambda)$ if for any $z_1, \cdots, z_n,$ the process
$$
M_t(z_1, \cdots, z_n):=\left(M \| \ti{w_t}^{-1} \right)(z_1, \cdots, z_n),\quad \tilde{w}_t:=\tilde{g}_t-\xi_t,\quad (t<T)
$$
(stopped when any $z_j$ or any $q_k$ is swallowed by the Loewner hull $\tilde{K}_t$) is a local martingale on the annulus SLE probability space.

\subsection{Main results} \label{Main result sub}
For a smooth vector field $v,$ let us denote by $\mathcal L^+_v$ (resp., $\mathcal L^-_v$) the $\C$-linear (resp., anti $\C$-linear) part of Lie derivative operator $\mathcal L_v$ with respect to $v,$ i.e., $\mathcal L_v^\pm = (\mathcal L_v \mp i \mathcal L_{iv})/2.$
We define a \textit{stress tensor} $A_{(b)}$ by
$$ A_{(b)}:= -\frac{1}{2}\, \pa \Phi_{(b)} \odot \pa \Phi_{(b)} +ib \, \pa^2 \Phi_{(b)},
$$
where $\odot$ denotes the Wick's product.
See Subsection~\ref{CCM} for more details.

We derive the following form of Ward's equation for both ER and Dirichlet boundary conditions.
\begin{thm} \label{Ward eq_main}
	For any string $\XX$ of fields in the OPE family $\FF_{(b)},$ we have
	$$
	2\E \left[A_{(b)}(\zeta) \XX \right]=\Big(\LL_{v_\zeta}^+ +\LL_{v_{\bar{\zeta}}}^- \Big)\E\left[\XX \right]+\pa_r \E \left[\XX \right],
	$$
	where all fields are evaluated in the identity chart of $\mathcal C_r$ and the Loewner vector field $v_{\zeta}$ is given by
$$
(v_\zeta \| \id_{\bar{\mathcal C}_r})(z)=H(r,\zeta-z).
$$
\end{thm}
To define the one-leg operator $\Psi$ or the modified multi-vertex fields (OPE exponentials) in general, it is convenient to introduce the formal (1-point) bosonic fields $\Phi_{(0)}^{\pm}$ in $D_r.$ Although they are not Fock space fields, they can be interpreted as the ``holomorphic part" and the ``anti-holomorphic part" of the GFF $\Phi_{(0)}$ in the following sense:
$\Phi_{(0)}=\Phiplus_{(0)}+\Phiminus_{(0)}$, $\Phiminus_{(0)}=\overline{\Phiplus_{(0)}}.$
Given divisors $\bfs{\sigma}=\sum_{j=1}^n \sigma_j \cdot z_j,$ $\bfs{\sigma_*}=\sum_{j=1}^n \sigma_{j*} \cdot z_j,$
we set
$$
\Phi_{(0)}[\bfs{{\sigma}},\bfs{{\sigma_*}}] := \sum_{j=1}^n \sigma_j\Phiplus_{(0)}(z_j)-\sigma_{*j}\Phiminus_{(0)}(z_j).
$$
Then $\Phi_{(0)}[\bfs{{\sigma}},\bfs{{\sigma_*}}]$ is a well-defined Fock space field if and only if the following neutrality condition $(\mathrm{NC}_0)$ holds:
$$\sum_{j=1}^n \left(\sigma_j+\sigma_{*j} \right)=0.$$
Given a double divisor $(\bfs{{\sigma}},\bfs{{\sigma_*}})$ for $z_j \in \CC_r $ satisying the neutrality condition $(\mathrm{NC}_0),$ we define the \textit{multi-vertex field} $\OO{[\bfs{\sigma},\bfs{\sigma_*}]}\equiv \OO_{(b)}{[\bfs{\sigma},\bfs{\sigma_*}]}$ by
\begin{equation*} \label{mvf_main}
 \OO_{(b)}{[\bfs{\sigma},\bfs{\sigma_*}]} = C_{(b)}{[\bfs{\sigma},\bfs{\sigma_*}]} e^{\odot i\Phi_{(0)}[\bfs{{\sigma}},\bfs{{\sigma_*}}]},
\end{equation*}
where $C_{(b)}{[\bfs{\sigma},\bfs{\sigma_*}]}$ is a differential (see its definition in Subsection~\ref{Conformal Subsec}) of conformal  dimensions $(\lambda_j,\lambda_{*j})$ at $z_j$ $(\lambda_j = \sigma_j^2 /2 -b\sigma_j, \lambda_{*j} = \sigma_{*j}^2 /2 -b\sigma_{*j})$ and the evaluation of $C_{(b)}{[\bfs{\sigma},\bfs{\sigma_*}]}$ in the identity chart of $\CC_r$ is given by
\begin{align*}
&\Theta'(0)^{\frac12\sum_j (\sigma_j^2+\sigma_{*j}^2)} \prod_j \Theta(z_j-\bar{z}_j )^{\sigma_j\sigma_{*j}}
\\
\times& \prod_{j<k} \Theta(z_j-z_k)^{\sigma_j\sigma_k}\Theta(\bar{z}_{j}-z_k)^{\sigma_{*j}\sigma_k} \Theta(z_j-\bar{z}_k)^{\sigma_j\sigma_{*k}}\Theta(\bar{z}_j-\bar{z}_k)^{\sigma_{*j}\sigma_{*k}}
\end{align*}
for CFT with ER boundary condition.
To obtain the evaluation for Dirichlet boundary condition, one needs to replace all $\Theta$ functions in the above formula with $\tilde{\Theta}.$

To connect CFT with SLE theory, we choose real parameters $a$ and $b$ in terms of SLE parameter $\kappa$ as
\begin{equation*} \label{a,b,kappa}
a=\sqrt{2/\kappa}, \qquad b=\sqrt{\kappa/8}-\sqrt{2/\kappa}.
\end{equation*}
Given a divisor $\bfs{\beta}= \sum_{j=1}^N \beta_j \cdot q_j$ satisfying $ a+\sum_{j=1}^N\beta_j =0,$
we define the one-leg operator $\Psi \equiv\Psi _{\bfs\beta}$ associated with $\bfs\beta$ by
$$
\Psi \equiv \Psi_{\bfs{\beta}}(p,{\bfs q}) := \begin{cases} \OO [ a\cdot z +{\bfs{\beta}}/{2} , {\bfs{\beta}}/{2} ] \,&\textrm{for ER boundary condition (b.c.),}
\vspace{0.5em}
\\
\hspace{0.1em}\OO [a\cdot z +\bfs{\beta}, \bfs{0}] &\textrm{for Dirichlet boundary condition (b.c.)}. \end{cases}
$$
Now we consider $\SLE(\kappa,\Lambda)$, where $\Lambda$ is set as
\begin{equation*} \label{Lambda_main}
\Lambda(r,p,\bfs{q}):=\kappa \frac{ \pa_{\xi}\big|_{\xi=p} \E \left[ \Psi(\xi,\bfs{q}) \right]}{\E \left[ \Psi(p,\bfs{q}) \right]}
\end{equation*}
for Dirichlet boundary condition. For the ER counterpart, certain renormalization procedure is required in the definition of $\Lambda$, see Subsection~\ref{one-leg subsec}.
Insertion of the one-leg operator produces an operator that changes the values of Fock space fields. See \cite[Section~2.3]{MR3052311} for its probabilistic counterpart: insertion of Wick's exponential $e^{\odot \Phi(f)}$ of the GFF in application to a test function $f$ results in the change of the law of $\Phi(f)$ in terms of the Green potential.
We now consider the following insertion procedure:
\begin{equation*} \label{modi cor_main}
 \wh{\E}[\mathcal X]:=\frac{\E[\Psi_{ \bfs{\beta} }(p , \bfs{q})\XX]}{ \E \left[ \Psi_{ \bfs{\beta}  }   (p, \bfs{q}) \right]}
\end{equation*}
for Dirichlet boundary condition. Again, we need a renormalization procedure for ER one.
The following theorem holds for both ER and Dirichlet boundary conditions.
\begin{thm} \label{main}
	For any string $\XX$ of fields in the OPE family $\FF_{(b)},$ a non-random field
	$M=\wh{\E}[\XX]$ is a martingale-observable for $\SLE(\kappa,\Lambda).$
\end{thm}

In Subsection~\ref{Screening_sub}, we focus on Lawler and Zhan's \textit{chordal type} annulus $\SLE(\kappa,\Lambda)$ with one force point $q$ starting from $p,$ where both of the marked points lie on the same boundary component. The drift function $\Lambda$ is defined in terms of the \textit{annulus SLE partition function} $Z \equiv Z(r,\cdot)$ by $\Lambda= \kappa Z'/Z.$ The partition function $Z$ satisfies the null-vector equation:
\begin{align}\label{Lawler Zhan null}
\pa_r Z & = \frac{\kappa}{2} Z'' + H\, Z' + h_{1,2}(\kappa)H'\, Z+C(r) Z, \qquad h_{1,2}(\kappa) := \frac{6-\kappa}{2\kappa},
\end{align}
where $C(r)$ is a constant depending only on the modular parameter $r$. For more details we refer the reader to \cite{MR2902453,lawler2011defining}.

To develop CFT for Lawler and Zhan's annulus SLE, we use the method of \textit{screening}. Due to the reversibility of annulus SLE$(\kappa,\Lambda)$ (at least for $\kappa \le 4$), the partition function (i.e., the correlator of corresponding one-leg operator) should have the same conformal dimensions at $p$ and $q.$
In the case of chordal SLE in the upper-half plane, it can be achieved by the \textit{effective one-leg operator} with the total charge $2b-a$ (including the background charge $2b$) placed at $q$ and with the charge $a$  placed at $p.$ For more details we refer the reader to \cite{kang2012radial}. However, this choice of partition function cannot be realized in the annulus because the neutrality conditions depend on the connectivity. See \cite{KM17} for the neutrality conditions on a compact Riemann surface of genus $g.$
Instead we consider
	$$\Psi(p,q):=C(\kappa) \oint_{\mathcal{P}(p,q)} \OO[a\cdot p +a\cdot q +s \cdot \zeta, \, \bfs{0}] \, d\zeta, \qquad (s = -2a)$$
for both ER and Dirichlet boundary conditions, where the integration contour $\mathcal{P}(p,q)$ stands for the Pochhammer contour entwining $p$ and $q$, see Figure~\ref{Pochh} in Subsection~\ref{Screening_sub}. Here $C(\kappa)$ is a universal constant depending only on $\kappa,$ see \eqref{scr normal const}.
With this choice of $s =-2a,$ the OPE exponential $\OO[a\cdot p +a\cdot z +s \cdot \zeta, \, \bfs{0}]$ is a $1$-differential with respect to the ``screening" variable $\zeta$ and it satisfies the neutrality condition $(\mathrm{NC}_0).$
Clearly, $\Psi$ has the same conformal dimensions both at $p$ and $q.$ Moreover we have the following expression of the partition function $Z\equiv Z(r,p-q):=\E\Psi(p,q)$ in terms of Jacobi's theta function:
$$
Z=\begin{cases}
\displaystyle C(\kappa) \, \Theta'(0)^{\frac{6}{\kappa}} \Theta(p-q)^{\frac{2}{\kappa}} \oint_{\mathcal{P}(p,q)}
\Theta(p-\zeta)^{-\frac{4}{\kappa}} \Theta(\zeta-q) ^{-\frac{4}{\kappa}} d\zeta &\text{for ER b.c.}
\vspace{0.5em}
\\
\displaystyle C(\kappa) \, \tilde{\Theta}'(0)^{\frac{6}{\kappa}} \tilde{\Theta}(p-q)^{\frac{2}{\kappa}}
\oint_{\mathcal{P}(p,q)} \tilde{\Theta}(p-\zeta)^{-\frac{4}{\kappa}} \tilde{\Theta}(\zeta-q) ^{-\frac{4}{\kappa}} d\zeta &\text{for Dirichlet b.c.}
\end{cases}
$$
We remark that for $\kappa >4$, this Pochhammer contour integral is simplified to the integration along the interval between $p,q$, see \eqref{Euler integral}. It can be shown by the contour deformation method as we explain in Subsection~\ref{Screening_sub}. With this choice of one-leg operator $\Psi(p,q)$, we construct a family of martingale-observables for Lawler and Zhan's chordal type $\SLE(\kappa,\Lambda)$.
\begin{thm} \label{main_SLE Z}
For each $\kappa >0$, the partition function $Z:=\E\,\Psi(\cdot,q)$ is a non-trivial (real-valued) solution of the null vector equation \eqref{Lawler Zhan null}.
 Moreover, for any string $\XX$ of fields in the OPE family $\FF_{(b)}$, a non-random field
	$$
	M=\wh{\E} \XX:= \frac{\E\Psi(p,q) \XX}{ \E\Psi(p,q) }
	$$
	is a martingale-observable for chordal type $\SLE(\kappa,\Lambda).$
\end{thm}

\medskip
\noindent\textbf{Organization of the paper.} In Section~\ref{CFT with ER b.c.} we review some of the standard notions in CFT and describe the conformal Fock space fields generated by central charge modifications of GFF with ER and Dirichlet boundary conditions. Section~\ref{WARD Section} is devoted to the study of Ward's equations (Theorem \ref{Ward eq_main}) in the annulus. In Section~\ref{SLE MOs ER} we relate CFT to SLE theory and present the proof of Theorem~\ref{main}. In the last subsection, we proceed with the study of null-vector equations and prove Theorem~\ref{main_SLE Z}.

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config: arxiv


TITLE: Poisson integral in $R^2$
QUESTION [3 upvotes]: This question arises out of confusion about some aspects of the Poisson integral in $R^2$, simplified here to a function f on the boundary of the unit circle $ \partial C$ and on C,
$$u(r,\phi) = \frac{1}{2\pi} \int_0^{2\pi} \frac{ f (\theta) (1-r^2) d\theta}{1-2r \cos(\theta-\phi)+r^2}$$
The intuition behind the derivation of the integral is described nicely here
and based on this sort of intuition I wondered whether there might be an approach that did not involve the use of "inversive" geometry.
The article cited mentions that we would typically be considering harmonic functions (temperature, e.g.) and that given an arbitrary piecewise continuous assignment of values to the boundary $\partial C$, "there always exists a harmonic function in R that takes on these values as the boundary is approached."
Well, suppose $f(\phi) = \phi^2  $? There is a discontinuity at $(r = 1, \phi = 0, 2 \pi )$. According to the article, if we are approaching $ \mathit{continuous }$ boundary points this is not an issue.
So for the question(s). If I guess--naively-- that the influence of a point on the boundary on a point inside the circle is inversely proportional to the square of the distance between the two points, I might approximate the integral as
$$ u \approx  \frac{\sum_{i=1}^n  \frac{f(\phi_i) }{(d_i)^2}}{\sum_{i=1}^n \frac{1}{(d_i)^2}}$$
This approximation is not great, but it yields reasonable values near the center of the circle.

Is there a straightforward Riemann-sum approximation of the integral along these lines? Or are we stuck with the geometry of the article? Is the partial agreement I get for my guess fortuitous?
We could assign pretty outrageous piecewise-continuous values to the boundary--do every one of these correspond to a theoretically possible (for example) heat distribution?

Thanks for any insights, edits.

REPLY [3 votes]: The Poisson kernel $$P(r,\theta,\phi)=\frac{ 1-r^2 }{1-2r \cos(\theta-\phi)+r^2}$$ can be written in the more intuitive form $$P(z,\zeta)=\frac{1-|z|^2}{|z-\zeta|^2}$$ where $z=re^{i\theta}$ and $\zeta=e^{i\phi}$. This form clearly shows both the inverse square relation (in the denominator) and penalty for being close to the boundary (in the numerator). Notice that $1-|z|^2=(1+|z|)(1-|z|)$ where the first factor does not change much: $1\le 1+|z|\le 2$ for all $z$ in the disk. Thus, the penalty for proximity to the boundary is essentially linear, $(1-|z|)$.
Your sum with squared distances ignores the numerator, which of course leads to poor results near the boundary. If you want a Riemann sum approximation, why not just take the Riemann sum of the original Poisson integral itself: that is, split $[0,2\pi]$ into subintervals and pick a point in each?
Concerning (2): Anything piecewise continuous will work as you expect as long as the boundary values are integrable. For example, the boundary values $1/\theta$ for $\theta\in (0,2\pi]$ will not give you any harmonic function inside in the disk: the integral will simply diverge everywhere. But for any integrable boundary values you get a harmonic function, which can be thought of as a stationary heat distribution (under the assumption that the temperature of the boundary is somehow maintained by an outside device).


TITLE: What are the main ideas needed to prove that only $92$ Johnson solids exist?
QUESTION [12 upvotes]: The Johnson solids are the strictly convex polyhedra made out of regular faces, excluding the vertex-transitive ones (which instead enter into the uniform category). Victor Zalgaller proved in 1969, in his paper Convex Polyhedra With Regular Faces, that only $92$ such polyhedra existed. I've read that he did this by a tedious computer search. I’m not aware of any other document that covers this proof.
The problem is, even after scouring the internet (and even some not fully legitimate sites) for about an hour, I've been unable to find a copy of his paper anywhere (except for this one, in Russian). And I have no idea of how I would replicate the argument myself. It doesn't even seem trivial that there is a way to reduce the problem to finitely many calculations. In fact, the existence of prisms and antiprisms seems to effectively contradict that intuition: what's stopping us from taping a bunch of equilateral triangles to a $50$-gonal prism and creating a valid solid, for example? [EDIT: Ted's answer already redirects to another source explaining this.]
I'm not asking for a full rundown of the proof, that would be too much to ask. My question is:
What are the main ideas of Zalgaller's (or any other's) enumeration of the Johnson solids?
Any accessible reference is welcome.

REPLY [0 votes]: I quote Robert R. Tupelo-Schneck:

The complete list [of Johnson polyhedra] was given by Norman Johnson in 1966 and proven complete by Victor Zalgaller in 1969. The proof proceeds by an exhaustive case-analysis over all ways to attach regular faces together which preserve convexity as well as the property of being noncomposite. A regular-faced polyhedron is noncomposite if there is no plane which divides the polyhedron into two regular-faced polyhedra. Zalgaller established that the noncomposite strictly convex regular-faced polyhedra were the prisms and antiprisms (excepting A4, the octahedron, which is composite) and a further 28 polyhedra. Zalgaller concluded by stating the theorem that all junctions of those polyhedra along entire faces which are strictly convex are in the list of 110 [Platonic, Archimedean, and Johnson solids], establishing the classification.

Actually, it doesn't seem like Zalgaller actually completed the details of the argument. Quoting A. V. Timofeenko:

The completeness of that list was stated in [Zalgaller 1967] on page 18 as a theorem; however, it was supplied with only a brief indication to a method of proof, namely, the listing of all possible adjoins of noncomposite polyhedra without conditional edges along entire faces.

Here, a noncomposite polyhedron is a Johnson solid that cannot divided into two of them by a plane. It only remains to see how the noncomposite polyhedra were found.


text:
\begin{definition}[Definition:Elementary Matrix Operation]
=== [[Definition:Elementary Row Operation|Elementary Row Operation]] ===
{{:Definition:Elementary Row Operation}}
=== [[Definition:Elementary Column Operation|Elementary Column Operation]] ===
{{:Definition:Elementary Column Operation}}
\end{definition}

config: wiki


TITLE: Calculation of two dimensional Fourier transform on a disk using FFT?
QUESTION [0 upvotes]: How to calculate the FFT of the function $f(x,y)$ defined on a disk ($\sqrt{x^2+y^2}<r$)?
It seems all published FFT code deal with 2-dimensional Fourier transform for functions defined on rectangular areas. Can the existing code, by some modification, be used to compute the FFT of functions on a disk?

REPLY [0 votes]: You would have to define first what the DFT on a disk is. The DFT can in general be defined for any group of translations. Take the group defined by two translation vectors in the plane and you get the DFT on the torus or unwrapped a rectangle or parallelogram.
I don't see a similar group action for the circle.


TITLE: Beautiful bent triangles
QUESTION [0 upvotes]: $x°, y°, z°$ are angles of tight rope triangle $ABC$. A killer bully bent its sides one degree each. So he got a concave hexagon with $(x-1)°, 181°, (y-1)°, 181°, (z-1)°, 181°$ as its angles. Prove that points of bending divided the sides of original triangle in the same ratio.
It surely does sound nice, but I struggled for a long time to solve it. I tried to use similarity and coordinates, but then realized sides can't be streched. I'm still trying, but completely stuck.

REPLY [0 votes]: Look at the following illustration, where angles are exaggerated:

The triangle $A'B'C'$ may be your original triangle, or a shrunken version thereof. In any case, it has the same interior angles as the original triangle $ABC$.
You have $\angle A'DB'=\angle B'EC'=\angle C'FA'=179°$. So $\angle B'A'D+\angle DB'A'=1°$. But at the same time, $\angle DA'F=(x-1)°$ so $\angle B'A'D+\angle FA'C'=1°$. If you combine them, you get $\angle DB'A'=\angle FA'C'$. Doing the same for other angles you find
$$\angle DB'A'=\angle EC'B'=\angle FA'C'\\
\angle B'A'D=\angle C'B'E=\angle A'C'F$$
So all three triangles $\triangle A'B'D,\triangle B'C'E,\triangle C'A'F$ are similar.
Since the argument above only uses angles, it doesn't matter whether the corners $ABC$ stay in their original positions and the rope is stretched, or whether the sum of the lengths for each original edge remains the same and the triangle as a whole gets scaled down a bit to obtain the new corners satisfying $\lvert A'D\rvert+\lvert DB'\rvert=\lvert AB\rvert$.
Of course, I started by assuming a triangle $A'B'C'$ similar to $ABC$. You might well challange that assumption. Having $A'B'$ similar to $AB$ is no big feat, since any line segment is similar to any other. So assume $A'B'$ to be all right, and then consider alternatives for $C'$. If $A',D,B'$ are known, then you also know the rays $A'F$ and $B'E$ but you might have some freedom of where on these rays you choose $E$ and $F$. More specifically you have two degrees of freedom, but you must choose them so that the edge sums for the two remaining original edges are satisfied again, which amounts to two constraints. Two degrees of freedom minus two constraints leaves you with no freedom at all.
The game changes if the corners of the triangle are just pegs, and the rope can move around them. In that case, the perimeter as a whole would have to remain constant, but for each edge the perimeter might change. I believe that in this case, your claim would no longer be true, since you'd have enough freedom to choose a different bending point for each edge.


TITLE: Solving a PDE for a function of 3 variables
QUESTION [0 upvotes]: I'm looking to solve the PDE below for a function $f(p,R,t)$.
$\frac{\delta f(p,R,t)}{\delta t}-D(R,t)\,p\,\frac{\delta f(p,R,t)}{\delta p}=0 $
I'm aware that the method of characteristics is supposed to be used to make progress, but I have no idea how to apply it in this case.

REPLY [1 votes]: $$\frac{\partial f(p,R,t)}{\partial t}-D(R,t)\,p\,\frac{\partial f(p,R,t)}{\partial p}=0 \tag 1$$
There is no partial differential with respect to $R$. Thus, $R$ can be considered as a parameter, or a dummy variable if you prefer. In a first step, we forget it in order to simplify the typing.
$$\frac{\partial f(p,t)}{\partial t}-D(t)\,p\,\frac{\partial f(p,t)}{\partial p}=0 \tag 2$$
The system of characteristic ODEs is :
$$\frac{dt}{1}=\frac{dp}{-D(t)p}=\frac{df}{0}$$
A first family of characteristic curves comes from $\frac{df}{0}=$differential of finite function of $t$ and $p$, which implies $df=0$
$$f=c_1$$
A second family of characteristic curves comes from $\frac{dt}{1}=\frac{dp}{-D(t)p}$ which is a separable ODE.
$$\ln|p|+\int D(t)dt=c_2$$
General solution of PDE $(2)$, expressed on the form of implicit equation :
$$\Phi\left(f\:,\:\left(\ln|p|+\int D(t)dt\right)\: \right)=0$$
where $\Phi$ is an arbitrary function of two variables.
Or equivalently on explicit form :
$$f(p,t)=F\left(\ln|p|+\int D(t)dt \right)$$
where $F$ is an arbitrary function.
Coming back to the PDE $(1)$ in re-introducing the dummy variable $R$ gives the general solution of PDE $(1)$ :
$$f(p,R,t)=F\left(\:\left(\ln|p|+\int D(R,t)dt \right)\:,\:R\:\right)$$
where $F$ is an arbitrary function of two variables.
No boundary and no initial condition are specified in the question of the OP. So, one cannot go further. If some viable boundary and initial conditions where specified one could try to determine a specific function $F$ of two variables fitting the specified conditions. Often this is the most difficult part of the task.


text:
\begin{document}

\title{A linear programming approach to inverse planning in Gamma Knife radiosurgery}

\author{J. Sjölund\footnote{Corresponding author. Email: \href{mailto:jens.sjolund@elekta.com}{jens.sjolund@elekta.com}
}, S. Riad, M. Hennix, H. Nordström}
\affiliation{Elekta Instrument AB, Kungstensgatan 18, Box 7593, SE-103 93 Stockholm, Sweden}

\date{\today}

\begin{abstract}

\textbf{Purpose:} Leksell Gamma Knife\textsuperscript{\textregistered} is a stereotactic radiosurgery system that allows fine-grained control of the delivered dose distribution. We describe a new inverse planning approach that both resolves shortcomings of earlier approaches and unlocks new capabilities.

\textbf{Methods:} We fix the isocenter positions and perform sector-duration optimization using linear programming, and study the effect of beam-on time penalization on the trade-off between beam-on time and plan quality. We also describe two techniques that reduce the problem size and thus further reduce the solution time: dualization and representative subsampling.

\textbf{Results:}
The beam-on time penalization reduces the beam-on time by a factor 2--3 compared with the na\"{i}ve alternative. Dualization and representative subsampling each leads to optimization time-savings by a factor 5--20. Overall, we find in a comparison with 75 clinical plans that we can always find plans with similar coverage and better selectivity and beam-on time. In 44 of these, we can even find a plan that also has better gradient index. On a standard GammaPlan workstation, the optimization times ranged from 2.3 to 26 s with a median time of 5.7 s.

\textbf{Conclusion:} We present a combination of techniques that enables sector-duration optimization in a clinically feasible time frame.

\end{abstract}

\keywords{Stereotactic radiosurgery, Leksell Gamma Knife\textsuperscript{\textregistered}, treatment planning, inverse planning, optimization}
\maketitle

\section{Introduction}
Stereotactic radiosurgery (SRS) is defined as the use of externally generated ionizing radiation to inactivate or eradicate defined targets, typically in the head or spine, without the need for a surgical incision \cite{Barnett2007}. Present-day neurosurgeons routinely use stereotactic radiosurgery for the management of a wide variety of brain disorders, including certain malignant and benign tumors \cite{Kondziolka1991,Kondziolka1999,Lunsford2005,Sheehan2005}, as well as cardiovascular and functional disorders within the brain \cite{Kondziolka2012, Lunsford1991, Maesawa2001, Regis2006}.

Leksell Gamma Knife\textsuperscript{\textregistered} (LGK) is a dedicated system for intracranial stereotactic radiosurgery. Its most recent incarnations, Perfexion\texttrademark and Icon\texttrademark, use 192 $^{60}$Co sources, each emitting gamma radiation. The radiation is collimated to create a focus where the radiation from every source converges. At the focus, both the radiation intensity and its gradient become very large. This makes it possible to deliver high radiation doses with minimal damage to surrounding healthy tissue.

The Perfexion\texttrademark \,and Icon\texttrademark \,systems enable two ways of tailoring the radiation dose according to the shape and size of the target. First, the patient can be precisely moved (robotically) in relation to the focus, effectively placing the focus in different \emph{isocenters}. Second, the radiation sources are arranged in eight, independently controlled, sectors.
Each sector can be in one of four different collimator states: the 4 mm, 8 mm or 16 mm or in the beam-off state. For each isocenter position and collimator configuration (i.e.~collimator size for each sector), the irradiation time can be specified. This composition is often referred to as a {\it shot}.

The large number of degrees of freedom allows sculpting of the dose distribution in unparalleled ways. At the same time, however, it is infeasible to explore them all by means of manual planning. Thus, an inverse planning method is required to make the full potential of  LGK clinically accessible. Inverse planning methods only require the user to specify what objectives to strive for, and then uses mathematical optimization to search for the best possible treatment plan according to these objectives.

Inverse planning was introduced in Leksell GammaPlan\textsuperscript{\textregistered} 10, and has since then become widely adopted. This inverse planner uses well-established metrics such as coverage, selectivity and gradient index at a pre-determined isodose level, together with a beam-on time (BOT) penalization. The optimization variables are the position of isocenters, collimator configurations and irradiation times (beam weights more precisely). Unfortunately, the optimization problem is inherently difficult (non-convex). The difficulty arises from two components: that the objectives use relative isodoses (instead of absolute doses) and that the positions of the shots can change. Besides, the direct use of relative isodoses makes it difficult to simultaneously manage multiple targets and to enforce criteria such as the maximum dose to an organs at risk (OAR).

Researchers have proposed optimization approaches that use absolute doses and shots that are allowed to move \cite{Ferris2000,Ferris2002,Ferris2003, Ghaffari2012mip, Lee2000, Vandewouw2016}. This typically results in a so-called mixed-integer problem, which remains non-convex. In practice, this implies that there is a compromise between computation time and the risk of ending up in a suboptimal solution. A consequence is that it is difficult to explore what trade-offs are achievable---especially in complicated cases with multiple conflicting objectives.

A remedy to most of these problems is to formulate a \emph{convex} optimization problem. Convexity is a highly desirable property that allows optimizations problems to be solved reliably and efficiently\cite{Boyd2004}.
One way to achieve convexity---which we also use---is to fix the isocenter positions and perform sector-duration optimization \cite{Ghobadi2012}. This approach is inspired by optimization formulations, in particular fluence map optimization, that are common in intensity-modulated radiation therapy (IMRT) \cite{Bortfeld1999,Romeijn2008}. In sector-duration optimization, the collimator configurations are not packaged into shots during the optimization, instead the irradiation times of every collimator in every sector are treated independently during the optimization. The irradiation times are converted into deliverable shots after a solution has been found. Despite its promises, methods using sector-duration optimization has, so far, lacked an efficient way of controlling BOTs \cite{Ghaffari2012,Ghaffari2017} and have been too computationally costly for widespread clinical use \cite{Ghobadi2012,Ghobadi2013,Vandewouw2016}. This is about to change.

We present the first sector-duration optimization that uses linear programming, which is possible thanks to a well-founded
BOT penalization. Linear programming has a long history in radiotherapy\cite{Ehrgott2008,Holder2005,Reemtsen2009,Rosen1991,Shepard1999}, dating back at least to 1968\cite{Bahr1968}. Many of the early approaches constrained the doses to predetermined intervals and used the remaining freedom to maximize an objective function such as the mean\cite{Rosen1991} or minimum\cite{Lodwick1999} dose in the target. But, in practice, this upfront specification of dose constraints often lead to infeasible problems. This motivated the use of elastic constraints\cite{Holder2003}, where one instead minimizes the mean or maximum of the (weighted) constraint violations. Our formulation belongs to this category, but with an additional term for BOT penalization.

Linear programming has also been used in radiosurgery before, specifically in two-phase approaches to inverse planning both for Gamma Knife radiosurgery\cite{Wu2003} and for robotic radiosurgery \cite{Schweikard1994,Schweikard1998,Kilby2010}. There, the decision variables are the weights of (deliverable) beams/shots that are generated in the first phase and optimized in the second.  In Gamma Knife radiosurgery, the sector-duration optimization has 24 times the number of decision variables of the corresponding two-phase procedure. The equivalent of a sector-duration optimization for a system with a multileaf collimator would include a decision variable for the left- and right positions of every leaf, resulting in a roughly hundredfold increase of decision variables. In this work, we present novel contributions that reduce the problem size of the sector-duration optimization while preserving the flexibility it offers.

In summary, our method
\begin{itemize}
\item Manages multiple targets;
\item Explicitly handles OARs;
\item Reproducibly finds the optimum given a fixed set of isocenter locations;
\item Runs in well under a minute;
\item Efficiently optimizes for short BOTs;
\item Allows hard constraints;
\item Allows the exploration of achievable trade-offs.
\end{itemize}

\section{Materials and Methods}
To achieve convexity, we divide the planning into three distinct phases: isocenter placement, optimization and sequencing. The isocenters chosen in the first phase remain fixed throughout the rest of the planning. In the optimization phase, we formulate an optimization problem where competing objectives are combined as a weighted sum. By changing weights it is straightforward to explore achievable trade-offs. Possible objectives include dose to target, sparing of OARs and a new---highly efficient---BOT penalization. During the optimization, times for each sector and collimator are allowed to vary independently. In the sequencing phase, these times are converted into deliverable shots.

\subsection{Isocenter placement}\label{sec:isocenters}
The first phase of the proposed inverse planner is to choose isocenter positions. These remain fixed throughout the subsequent phases, defining the search space of the optimization. The main objective of this phase is thus to provide enough freedom to find a high quality plan; the only harm in including extra isocenter positions is that it will take longer time to solve the optimization problem.

Although algorithms for automatic isocenter placement\citep{Wagner2000, John2005, Ghobadi2012, Bourland2000, Doudareva2015,Ferris2003} certainly have a role to play, we do not consider them to be the main focus of this work. Consequently, to remove this source of variability, we will reuse the isocenter positions from the corresponding (manual) reference plan in all examples that follow.

\subsection{Optimization}
As we will describe below, a major difference between the inverse planner we propose and previous approaches based on sector-duration optimization\cite{Ghaffari2012,Ghaffari2017,Ghobadi2012,Ghobadi2013,Vandewouw2016} is that it uses \emph{linear programming}. Such optimization problems are well-behaved and well-studied \cite{Bertsimas1997,Boyd2004,Nocedal2006}.

We recognize that there are multiple, possibly conflicting, objectives that are desirable. However, our working assumption is that the exact priority among these should be provided by the user. Consequently, we foresee that a number of options will be evaluated in the course of optimizing the treatment plan. This indeed reflects how it works presently, but the intention is that the proposed inverse planner will elucidate the achievable trade-offs. Formally, suppose we have $m$ cost functions, $\vecf=(f_1 (\vecx),\ldots,f_m (\vecx))^t$, representing e.g.~target dose, dose to an OAR and a penalization of BOT. This can be conveniently rearranged into a single cost function by forming a weighted sum, i.e.~by scalar multiplication with the weights $\vecw=(w_1,\ldots,w_m )^t$, where each weight quantifies the importance of the corresponding part of the cost function. We will now describe how we have defined the different parts of the optimization problem so that, taken together, it can be expressed as a linear programming problem.

\subsubsection{Dose-based objectives}
Given a fixed set of isocenter positions, our optimization variables are the irradiation times $\tOpt_{isc}$ corresponding to every isocenter $i$, collimator state $c$ and sector $s$. The dose $D$ at a position $r$ is linear in terms of these irradiation times,
\begin{equation}
D(r, t_{isc}) = \sum_{i=1}^{N_{iso}}\sum_{s=1}^{8}\sum_{c=1}^{3}\Phi_{isc}(r)\tOpt_{isc}.
\end{equation}
Often, we are interested in the doses $D_n$ given to a discrete set of voxels at positions $r_n$, where $n=1,\ldots, N$. Then, after appropriate rearrangements, we may evaluate the dose as a matrix-vector multiplication: $D_n=(\Phi t)_n$. Thus, the $(N\times 24N_\text{iso})$-matrix $\Phi$, which we refer to as the dose rate kernel, maps irradiation times to doses.

In regions of interest, each voxel can be assigned dose-based objectives that reflect the amount of underdosage or overdosage it receives. Typically, the regions of interest include all targets, a volume of healthy tissue surrounding each target \cite{Ferris2000,Ghobadi2012} and clinically relevant OARs. We express the dose-based objectives using one or several hinge functions,
\begin{equation}
\left(D_n-\hat{D}_n\right)_+ = \text{max}\left(D_n-\hat{D}_n, 0\right),\label{eq:hinge}
\end{equation}
where $D_n$ is the dose in the voxel in question and $\hat{D}_n$ is a reference dose. This function, or its square, is commonly used as a dose-based objective\cite{Ghobadi2012,Romeijn2003}. In principle, every voxel could have a unique dose-based objective, which makes it possible to, for instance, perform full-fledged dose painting by numbers \cite{Alber2003, Bentzen2005, Ling2000}. By introducing auxiliary variables, piecewise linear convex functions such as the hinge function above, can be recast as linear programming problems, cf. appendix \ref{appendix:explicit_primal}.

\subsubsection{Beam-on time penalization}
To control the treatment time, we use a highly efficient penalization function which we refer to as the idealized beam-on time (iBOT) \cite{Ghaffari2012, Ghaffari2017}. It is defined as
\begin{equation}
\Theta(t) = \sum_{i=1}^{N_\text{iso}}\underset{s}{\max}\sum_{c=1}^3 t_{isc},
\end{equation}
where $N_\text{iso}$ is the number of isocenters; $s$ and $c$, respectively, correspond to the 8 sectors and 3 collimators. We emphasize that iBOT is \emph{not} an $L_p$-norm, nor does it promote sparsity---in fact, it is quite the opposite: it encourages the total BOT of each sector to be equally long, which is advantageous since they can irradiate simultaneously.
By introducing auxiliary variables, the iBOT function can be expressed using linear programming \cite{Svensson2014}, cf. appendix \ref{appendix:explicit_primal}.

\subsubsection{Constraints}
In a linear programming problem it is possible to include {\it hard constraints}, i.e.~conditions that the solution must fulfill. Physics only dictates one such hard constraint, namely that all times must be non-negative, $t\geq0$, and therefore this is the only hard constraint that is strictly necessary---exempting the ``artificial'' constraints coming from the auxiliary variables. One can, optionally, include hard constraints on doses in various regions.

\subsubsection{Normal tissue sparing}
We promote normal tissue sparing by encompassing the target within two non-overlapping thick ``shells'' shaped according to the target surface. By penalizing high doses inside the inner and outer shells we can control selectivity and gradient index (cf. section \ref{sec:metrics}), respectively. These two shells may seem similar to the widely used concept of margins\cite{ICRU2010}, but unlike margins---which have well-defined clinical meanings---our shells are merely used to steer the optimization. We construct the inner and outer shells as geometric expansions, defined via the Euclidean distance transform, of the target. The inner shell is expanded from the target surface until its total volume is half that of the target volume. The outer shell is expanded from the outer surface of the inner shell until its volume is twice that of the target.

\subsubsection{Illustrative example---primal formulation}
We will base our exposition on a minimal, yet illustrative, example of the optimization problem when there is a single target and an OAR where we want the limit the maximum dose to at most $D_{\rm O}$. In this case, the cost function has four components that control the target dose, selectivity,
gradient index and BOT, respectively. We denote the prescription dose to the target $D_{\rm T}$ and the dose thresholds for selectivity and
gradient index by $D_\textrm{S}$ and $D_\textrm{G}$, respectively.  We thus arrive at the  optimization problem
\begin{equation}\label{eq:costfcn}
\begin{aligned}
& \underset{t}{\text{minimize}}
& & \frac{w_{\rm T}}{D_{\rm T} N_{\rm T}}\sum_{i=1}^{N_{\rm T}} (D_{\rm T}-(\Phi_{\rm T} \tOpt)_i)_++ \frac{w_{\rm S}}{D_{\rm S} N_{\rm S}}\sum_{i=1}^{N_{\rm S}}((\Phi_{\rm S} \tOpt)_i-D_{\rm S})_+ \\
&&&+\frac{w_{\rm G}}{D_{\rm G} N_{\rm G}} \sum_{i=1}^{N_{\rm G}}((\Phi_{\rm G} \tOpt)_i-D_{\rm G})_++\frac{w_{\rm BOT}}{D_\textrm{T}/\varphi_\text{cal}}\Theta(t)\\%\sum_{i=1}^{N_{\rm iso}} \max_s\sum_{c=1}^3 \tOpt_{isc} \\
& \text{subject to}& & \Phi_{\rm O} \tOpt\leq D_{\rm O}\\
&&& \tOpt\geq 0,
\end{aligned}
\end{equation}
where $\varphi_\text{cal}$ is the calibration dose rate,  $N_\alpha$ is the number of voxels in the structures $\alpha\in \{\textrm{T,\,S,\,G,\,O}\}$ and the weights $w_\alpha$ govern the relative importance of each term. Typically, we let $D_{\rm S} = D_{\rm T}$ to promote selectivity and $D_{\rm G} = D_{\rm T}/2$  to reduce the gradient index. In appendix \ref{appendix:explicit_primal}, we give the explicit representation of the problem in Eq.~\eqref{eq:costfcn} as a linear programming problem on standard form.

The example given here can easily be extended in several ways. For instance, additional structures could be incorporated as additional terms in the cost function, and homogeneous (target) dose distributions could be promoted by including a hinge function that penalizes overdosage.

\subsubsection{Representative subsampling}
The introduction of one auxiliary variable for every voxel in a relevant structure vastly increases the size of the optimization problem. In realistic cases, there could be, say, $10^5$ variables. Na\"ively solving the optimization problem thus becomes time-consuming at best, but it could even be impossible due to memory limitations. Thus, it is essential to reduce the size of the problem.

As a first step, we propose to use an approximation we refer to as \emph{representative subsampling}. Representative subsampling is based on the observation that the dose tends to vary quite smoothly from one position to another. True enough, there are regions like the penumbra region of the Gamma Knife (and flattening filter-free accelerators), that may exhibit stronger variability than others, but for practical purposes it is redundant to take the dose in every voxel into account during the optimization. We exploit this realization by sampling only a representative fraction of the voxels in each structure that we use in the optimization.

Formally, we may understand this from the observation that most of the criteria involving a structure $\Omega$ can be phrased as an integral
\begin{equation}
J = \int_\Omega f\left(r,D(r,t))\right)dr\,,\label{eq:integral}
\end{equation}
where $f$ is a function that assigns a cost to the dose, $D(r,t)=\Phi(r)t$, at position $r$. For instance, a dose-based hinge functions, as in equation \eqref{eq:hinge}, corresponds to $f(r,D(r,t))=\left(D(r,t)-\hat{D}(r)\right)_+$ and a hard constraint on the maximum dose corresponds to
\begin{equation}
f(r,D(r,t)) = \begin{cases}
0&\text{if }D(r,t)\leq \hat{D}(r),\\
\infty&\text{otherwise}.
\end{cases}
\end{equation}
If we discretize the integral \eqref{eq:integral} uniformly (i.e. at the sampling points) we recover the conventional expression
\begin{equation}
J\approx \frac{V(\Omega)}{N}\sum_{n=1}^N f(r_n, \Phi(r_n)t),
\end{equation}
where $V(\cdot)$ is the volume operator. However, it is well-known that this scales poorly with dimension---the number of samples required to reach a given precision increases exponentially with the dimension. To mitigate this, researchers have proposed to subsample voxels on a regular grid\cite{Ghaffari2012mip} or to aggregate voxels into clusters\cite{Scherrer2005,Ungun2018}. Another option is to use a stochastic gradient descent algorithm that resamples voxels at each iteration\cite{Martin2007}, but stochastic gradient descent is often ineffective on constrained problems, since every iteration requires a projection onto the feasible set \cite{Wang2016}.

Randomness is powerful, though. Instead of the deterministic sampling schemes mentioned above, we build upon the well-established efficiency of Monte Carlo integration to approximate Eq. \eqref{eq:integral} by sampling positions uniformly at random in the structure. In addition, because we want to subsample also when using minimum and maximum dose constraints in the target or maximum dose constraints in organs at risk, we include a separate term corresponding to positions sampled at random on the tessellated surface of the volume.

\subsubsection{Dualization}
The introduction of auxiliary variables in the primal formulation increases the number of variables---often severalfold. However, as can be appreciated from the explicit formulation in appendix \ref{appendix:explicit_primal}, the resulting matrix is highly structured; it could e.g.~be decomposed as the sum of a low-rank matrix and a sparse matrix. Exploiting this structure makes it possible to reduce computation times drastically. We have found that dualization reduces the computation time by a factor 5--20 depending on the features of the problem. Since strong duality holds for linear programming problems, the primal and dual problems are equivalent.

In appendix \ref{appendix:explicit_dual} we revisit our earlier example and give the explicit formulas for the corresponding dual problem. Importantly, however, the constraints introduced when rewriting the hinge function using auxiliary variables become trivial thanks to the dualization. The resulting size reduction leads to dramatic performance improvements when solving the dual compared to the original, primal, problem. Moreover, the computational gain due to dualization is entirely complementary to that of representative subsampling.

\subsection{Sequencing algorithm for shot composition}
For the Gamma Knife system to deliver the treatment composite shots are required, i.e.~collimator and sector configurations for each shot. The conversion from irradiation times for each isocenter position $\tOpt_{isc}$ to such shots is referred to as sequencing. The sequencing can be done in several different ways, and although we will not go into details here \citep{nordstrom2015sequencing}, it is worth noting that the result of the optimization will often lead to multiple shots in the same isocenter position.\\

\begin{figure}[ht]
\centering
\includegraphics[scale=0.6]{TV_PIV}
\caption{The target volume (TV) and planning isodose volume (PIV)}\label{fig:TV}
\end{figure}
\subsection{Evaluation}\label{sec:metrics}
In Fig.~\ref{fig:TV}, we define the target volume (TV) and the planning isodose volume (PIV), which are used, together with the volume operator $V(\cdot)$, to define the common radiosurgery metrics\cite{Torrens2014}:
\begin{eqnarray}
\textrm{Coverage:} & \hspace{4 mm} C = \frac{V(PIV \cap TV)}{V(TV)}\,, \\
\textrm{Selectivity:} & \hspace{4 mm} S = \frac{V(PIV \cap TV)}{V(PIV)}\,, \\
\textrm{Gradient index:} &\hspace{4 mm} GI = \frac{V(PIV_{ISO/2})}{V(PIV_{ISO})}\,.
\end{eqnarray}
In words, coverage is the fraction of the target volume that receives at least as high dose as prescribed, selectivity is the fraction of the planning isodose volume that encompasses the target volume, and gradient index describes the dose fall-off by the ratio of the volume that receives at least half the prescription dose to the planning isodose volume.
In addition to these metrics, plan quality is often evaluated based on the Paddick conformity index (PCI)\citep{Paddick2000}, which is defined as:
\begin{eqnarray}
\textrm{Paddick conformity index:} & \hspace{4 mm} PCI = C\cdot S.\label{eq:PI}
\end{eqnarray}

\section{Results}
We have evaluated both the overall performance of the proposed inverse planner as well as studied the novel components---the beam-on time penalization and the subsampling scheme---in isolation. We begin by reporting on the beam-on time penalization and the subsampling scheme. Then, we describe the overall performance on a set of 75 clinically acceptable reference plans. Finally, we consider one case in more detail. For this case, we illustrate the achievable trade-offs and show how to perform dose painting or create a homogeneous dose distribution.

\subsection{Beam-on time penalization}\label{sec:BOT}
To show the efficiency of the iBOT penalization, we compare with replacing the term $\Theta(t)$ in  Eq.~\eqref{eq:costfcn} with a simple summation over all times,
\begin{equation}
\Theta_\text{simple}(t) = \sum_{i=1}^{N_\text{iso}}\sum_{s=1}^8\sum_{c=1}^3 t_{isc},
\end{equation}
which we refer to as simple BOT (sBOT). The rest of the objective function in  Eq.~\eqref{eq:costfcn} is left unchanged.  We make the comparison on three different cases: one small acoustic neuroma, one medium sized acoustic neuroma and one irregular meningioma. By varying the weight of the (selectivity promoting) inner shell against the BOT penalization while keeping the other weights fixed, we obtain plans with different plan quality and BOTs. To get adequate statistics, we ran 1000 optimizations for each case and choice of penalization.
Plans with the same Paddick index and gradient index to within $\pm$ 1 $\%$  were tallied and the average BOT calculated for each bin. The bins were chosen to have Paddick and gradient indices within $\pm$ 5 \% of the clinical plan. Table \ref{tab:sBOT} summarizes the mean and standard deviation of the average BOT with the iBOT penalization and the average BOT with the sBOT penalization, as well as the mean and standard deviation of the ratios computed for each bin.
\begin{center}
\renewcommand\tabcolsep{6pt}
\begin{table}[ht]
    \caption{BOT (min) at 3 Gy/min when using different penalization terms}\label{tab:sBOT}
    \begin{tabular}{ l  c  c  c  }
    \hline\hline
     & Small acoustic neuroma & Medium acoustic neuroma & Irregular meningioma \\ \hline
    iBOT& 9.3 (2.6) & 21.6 (1.5) & 110.8 (3.1)  \\
    sBOT & 16.8 (1.9) & 75.2 (4.4) & 303.2 (11.6) \\
    iBOT/sBOT & 0.55 (0.16) & 0.29 (0.026) & 0.37 (0.029) \\
    Optimization times iBOT (s) & 0.135 & 3.2 & 3.3 \\
    Optimization times sBOT (s) & 0.125 & 3.4 & 3.5 \\
    \hline\hline
    \end{tabular}
    \end{table}
\end{center}
Clearly, iBOT results in plans of equal quality but markedly shorter BOT than sBOT, in particular for the two larger targets. The optimization times are practically equal.

 Figure~\ref{fig:iBOT_vs_BOT2} gives an overview of all the simulations for the small acoustic neuroma, for which the BOT reduction was the smallest (but still almost a factor of 2).
\begin{figure}
 \centering
 \subfigure[Paddick index vs. BOT]{
\label{fig:iBOT_vs_BOT1}
 \includegraphics[width = 0.5\textwidth]{G08PIBOT.pdf}}
 ~
 \subfigure[Gradient index vs. BOT]{
 \label{fig:GIvsBOT}
\includegraphics[width = 0.5\textwidth]{G08GIBOT.pdf}}
 \caption{Comparison of the two BOT penalization terms for the small acoustic neuroma case.}\label{fig:iBOT_vs_BOT2}
 \end{figure}
Figure \ref{fig:BOT_tradeoff} shows how the BOT, Paddick index and gradient index depend on the iBOT weight. Even though the Paddick index may appear independent of the iBOT in the small acoustic neuroma case (middle left), recall from Eq. \eqref{eq:PI} that it is the product of coverage and selectivity. Separately these are not constant as a function of BOT; a shorter BOT is obtained by increasing the total dose rate, which for Gamma Knife is done by using larger collimators and more sectors simultaneously. This smoothens the dose distribution so that coverage increases at the expense of selectivity and gradient index.
\begin{figure}
 \centering
 \subfigure{
 \includegraphics[width = 0.3\textwidth]{G08_BOT.pdf}}
 \subfigure{
 \includegraphics[width = 0.3\textwidth]{G46_BOT.pdf}}
 \subfigure{
 \includegraphics[width = 0.3\textwidth]{G42_BOT.pdf}}
 \\
 \subfigure{
\includegraphics[width = 0.3\textwidth]{G08_PI.pdf}}
\subfigure{
\includegraphics[width = 0.3\textwidth]{G46_PI.pdf}}
\subfigure{
\includegraphics[width = 0.3\textwidth]{G42_PI.pdf}}
\\
\subfigure{
\includegraphics[width = 0.3\textwidth]{G08_GI.pdf}}
\subfigure{
\includegraphics[width = 0.3\textwidth]{G46_GI.pdf}}
\subfigure{
\includegraphics[width = 0.3\textwidth]{G42_GI.pdf}}
 \caption{Dependence of BOT (top), Paddick index (middle) and gradient index (bottom) on the iBOT penalization. The columns correspond to different cases: small acoustic neuroma (left), medium acoustic neuroma (center) and irregular meningioma (right).}\label{fig:BOT_tradeoff}
 \end{figure}

\subsection{Subsampling}
Since we subsample stochastically, the resulting plan metrics also become stochastic. We do, however, want to choose the subsampling fractions such that the statistical fluctuations remain reasonably small. Here, we require the standard deviation of both coverage and selectivity to be below 1\%.
Furthermore, we want to ensure that our sampling strategy, which samples both interior and surface points, performs at least comparably to sampling only in the interior.

We compare the two strategies on the three cases studied in Sec.~\ref{sec:BOT} and an arteriovenous malformation case (AVM). Before sampling, the number of target voxels are 6119 (small acoustic neuroma), 12078 (medium acoustic neuroma), 27765 (meningioma) and 30326 (AVM). We adjust the voxel size for the interior point sampling so that the total number of points in the target is the same for both sampling methods.
We performed 100 runs for each of seven subsampling fractions. Figure~\ref{fig:sampling} presents the resulting standard deviation of coverage and selectivity as a function of the percentage of the total number of voxels, i.e. the voxels belonging to all structures in the problem. For each sampling method we used a fixed weight setting ($(w_{\rm T},w_{\rm S},w_{\rm G},w_{\rm BOT})=(1.0,0.15, 0.15, 0.15)$ for our sampling method and $(w_{\rm T},w_{\rm S},w_{\rm G},w_{\rm BOT})=(1.0,0.025, 0.025, 0.25)$ for interior sampling only) that rendered clinically acceptable plans with metrics within 1\% of each other.
From Fig.~\ref{fig:sampling}, we conclude that our sampling strategy performs  comparably to the interior point sampling for both coverage and selectivity. Also, we see that the standard deviation of both metrics is below 1\% when the subsampling fraction is at least 10\%. The value of the metrics, and thus the standard deviation, is computed on the original grid. For small sampling fractions, the mean will deviate from the true value. However, for the sampling sizes we are interested in, i.e. about 10\%, this is typically not an issue.
\begin{figure}[ht!]
\centering
\subfigure[Coverage]{
\label{fig:first}
\includegraphics[width = 0.47\textwidth]{coverage_forPaper}}
~
\subfigure[Selectivity]{
\label{fig:second}
\includegraphics[width = 0.5\textwidth]{selectivity_forPaper}}
\caption{The standard deviation of coverage and selectivity for the two sampling methods, interior and surface ($*$) and interior only ($\circ$) for the small acoustic neuroma (red), the medium acoustic neuroma (purple), the irregular meningioma (yellow) and the arteriovenous malformation (blue).}\label{fig:sampling}
\end{figure}

As illustrated in Fig.~\ref{fig:opt_times}, subsampling shortens the optimization time. Evidently, the dependence between the optimization time and the subsampling fraction is approximately linear. In the cases considered above, asubsampling of 10\% shortens the optimization time by a factor 8--22 compared to using all initial sampling points.
\begin{figure}
\includegraphics[width = 0.6\textwidth]{optimization_times}
\caption{The optimization time as a function of subsampling fraction for small acoustic neuroma (red), the medium acoustic neuroma(purple), the irregular meningioma (yellow) and the arteriovenous malformation (blue).}\label{fig:opt_times}
\end{figure}
\subsection{Overall performance}
We evaluate the overall performance by comparing the plan metrics for optimized plans with manual forward plans for 75 clinical cases (the majority of Gamma Knife users still do forward planning). The clinical cases are all single target cases: single metastases, acoustic neuromas and meningiomas. The range in tumor size is 0.6--11.7 ${\rm cm}^3$ and the planning dose range is 12--24 Gy.  In 18 of these cases there are organs at risk (e.g. brainstem, cochlea, optic nerves) so, to make a fair comparison, the maximum doses to the OARs in the manual plans are used as hard constraints in the optimization. Furthermore, all the manual plans have 98--100\% coverage, so the weights were chosen to always satisfy this criteria. To compare the manual and optimized plans, we have to find weight settings for each optimized plan that give a similar trade-off between clinical objectives as the corresponding manual plan. For simplicity, we employed the following strategy: first, we created a range of plans with different characteristics by performing 101 optimizations for each case, with randomly sampled weights for the inner ring and BOT penalization; then we selected, for each case, the optimized plan with the best gradient index among those with both higher selectivity and shorter BOT than the corresponding manual plan. The resulting plan metrics are shown in figure~\ref{fig:75_cases_figure}. In summary, for all of the 75 cases we could find optimized plans with simultaneously higher selectivity and shorter BOT than the manual plans, and in 44 cases these plans also had better gradient index than the manual plans. In other words, we found plans that dominate the manual ones in almost 60\% of the cases.

To solve the optimization problems, we used the open-source solver Glop\cite{Glop} with default settings. The optimization times ranged from 2.3 to 26 s with a median time of 5.7 s on a standard GammaPlan workstation (a HP Z640 with 32 Gb RAM and 12 cores running at 2.9 MHz). Figure~\ref{fig:75_cases_sizeB_NZ} shows how the optimization time depends on the number of non-zero elements in the constraint matrix $B$ (cf. appendix \ref{appendix:explicit_dual}). We conclude that there is a linear trend, but that the variation appears larger for small cases.

\begin{figure}
\centering
\subfigure[Selectivity for all the cases]{
\includegraphics[width = 0.65\textwidth]{75_cases_sel.pdf}}
\\
\subfigure[BOT for all the cases]{
\includegraphics[width = 0.65\textwidth]{75_cases_bot.pdf}}
\\
\subfigure[Gradient index for all the cases]{
\includegraphics[width = 0.65\textwidth]{75_cases_gi.pdf}}
\caption{Metrics for the optimized plans compared to the forward plans. The cases are sorted by the difference in gradient index.}\label{fig:75_cases_figure}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[scale=0.3]{75_cases_sizeB_NZ_opt_times.pdf}
\caption{Optimization time as a function of the number of non-zero elements in B for the 75 clinical cases. The times shown are averages over the 101 runs made for each case.}\label{fig:75_cases_sizeB_NZ}
\end{figure}

\subsection{Detailed case study}\label{sec:GFRAME41}
As an example, we consider a large left-sided cavernous sinus meningioma, with a volume of  $16.2\,\textrm{cm}^3$ and with three adjacent OARs: the chiasm, the left optic nerve and the left optic tract. The prescription dose to target is 15 Gy, the dose in the inner shell encompassing the target is penalized if the dose exceeds 15 Gy. In the outer shell we penalize doses exceeding 7.5 Gy. The maximal allowed dose to all three OARs is 8 Gy, which is enforced by hard constraints. In Tab.~\ref{tab:GFRAME41}, we present two optimized plans, one promoting selectivity and one promoting short BOT, together with the clinical plan for comparison. Figure~\ref{fig:plans} shows snapshots and dose-volume histograms from Leksell GammaPlan\textsuperscript{\textregistered} for the three different plans. In the plan where short BOT is promoted, the BOT is almost halved at the expense of a moderate decrease in selectivity compared to the second plan. However, we consider both plans clinically acceptable.

\begin{center}
\renewcommand\tabcolsep{6pt}
\begin{table}[ht!]
    \caption{Plan metrics for the reference plan and two optimized plans}\label{tab:GFRAME41}
    \begin{tabular}{ l  c  c  c  }
    \hline\hline
     & Reference plan & Promoting selectivity & Promoting BOT \\ \hline
    Coverage& 0.95 & 0.95 & 0.96  \\
    Selectivity & 0.88 & \textbf{0.91} & 0.78 \\
    Gradient index & 2.73 & 2.79 & 2.71 \\
    BOT (min) at 3 Gy/min & 89 & 135 & \textbf{41} \\
    Max dose (Lt Optic nerve) &8.95& 8.0 & 7.2 \\
    Max dose (Lt Optic tract) & 7.3 & 8.0 & 7.2  \\
    Max dose (Chiasm) & 7.3& 8.0 & 7.2\\
    Planning isodose & 46 & 44 & 53  \\

    \hline\hline
    \end{tabular}
    \end{table}
\end{center}

\begin{figure}
\centering
\subfigure[Reference]{
\includegraphics[width = 0.33\textwidth]{Reference.pdf}}
\qquad
\subfigure[Reference]{
\includegraphics[width = 0.55\textwidth]{ReferencePlan.pdf}}
\qquad
\subfigure[Promoting selectivity]{
\label{fig:first}
\includegraphics[width = 0.33\textwidth]{pq.pdf}}
\qquad
\subfigure[Promoting selectivity]{

\includegraphics[width = 0.55\textwidth]{PromoteSelectivity.pdf}}
\qquad
\centering
\subfigure[Promoting BOT]{
\label{fig:first}
\includegraphics[width = 0.33\textwidth]{BOT.pdf}}
\qquad
\subfigure[Promoting BOT]{

\includegraphics[width = 0.55\textwidth]{PromoteBOT.pdf}}
\caption{Snapshots and DVHs of the reference plan, the plan promoting selectivity and the plan promoting BOT for the meningioma with adjacent OARs. The DVH for the target is in red and the three OARs in magenta, light blue and pink.}\label{fig:plans}
\end{figure}

\subsubsection{Dose painting and homogeneous plans}
The flexibility of the proposed inverse planner opens up new possibilities for treatment planning. Here, we will exemplify both dose painting and how to create plans with a homogeneous dose distribution in the target.

Dose painting is achievable since both weights and prescription doses can be modified on a voxel-by-voxel basis. To illustrate this, we introduce a hotspot, with a volume of $1.2\, \textrm{cm}^3$, in the center of the target. The dose prescribed to the hotspot is twice the dose prescribed to the rest of the structure, i.e.~30 Gy. In the present example, we get a hotspot coverage of 99~\%.

It turns out that the iBOT term will in many cases favor fairly homogeneous plans. However, such plans can be further promoted by penalizing overdosage of the target. We thus penalize doses exceeding $15/0.85$ Gy to get a planning isodose of at least 85\%, which is very difficult using forward planning. In Tab.~\ref{tab:hotspot}, we present an example of a plan for the hotspot case and one plan with a homogeneous dose distribution. Overall, we consider both plans acceptable, although naturally the additional requirements result in some form of trade-off. The hotspot case has similar plan quality but longer BOT than the clinically acceptable reference plan presented in Tab.~\ref{tab:GFRAME41}. The homogeneous plan, on the other hand, trades off selectivity and we need to loosen the hard constraint on dose to OARs to achieve homogeneity. In Fig.~\ref{fig:specialplans}, we present snapshots of one plan with a homogeneous dose distribution and one plan with a hotspot.
\begin{center}
\begin{table}[ht!]
\renewcommand\tabcolsep{6pt}
\caption{One plan with a hotspot and one homogeneous plan}\label{tab:hotspot}
    \begin{tabular}{l  c  c  }
    \hline\hline
     & Hotspot\: & Homogeneous plan \\ \hline
    Coverage& 0.97& 0.95  \\
    Selectivity & 0.85&  0.87 \\
    Gradient index & 2.9& 2.8  \\
    BOT (min) at 3 Gy/min & 136 & 152\\
    Max dose target (Gy) &  38.5& 17.7 \\
    Max dose Lt Optic nerve (Gy) & 8.0 & 8.0 \\
    Max dose Lt Optic tract (Gy) & 8.0 & 8.0  \\
    Max dose Chiasm (Gy) & 8.0 & 8.0\\
    Planning isodose &  40&  85 \\
    \hline\hline
    \end{tabular}
    \end{table}
\end{center}

\begin{figure}
\centering
\subfigure[Homogeneous plan]{

\includegraphics[width = 0.7\textwidth]{GFRAME41_Homogeneous_cropped}}
\qquad
\subfigure[Plan with hotspot]{

\includegraphics[width = 0.7\textwidth]{GFRAME41_hotspot}}
\caption{Snapshots from one plan with homogeneous dose distribution and one with a hotspot}\label{fig:specialplans}
\end{figure}

\section{Discussion}
In this work, we have described a linear programming approach to Gamma Knife radiosurgery based on the division of the planning into three distinct phases: isocenter placement, optimization and sequencing. Our main focus has, however, been the optimization phase. We have shown that even without changing the isocenter positions, our optimization approach can find plans that dominate manual forward plans in almost 60\% of the cases investigated. Reusing the isocenter positions in this way clarifies the improvement that results directly from the optimization, but it is---of course---not how our approach would eventually be deployed. A natural next step is thus to investigate methods for isocenter placement with an emphasis on the interplay with subsequent phases.

Our optimization problem, with a cost function that is a weighted sum of multiple competing objectives, is fundamentally a multi-criteria optimization problem. A useful concept in multi-criteria optimization is that of the Pareto surface. Simply put, the Pareto surface is the set of solutions in which you cannot improve one objective without impairing another. This means one should never be satisfied with a solution that is not on the Pareto surface (Pareto optimal) unless other considerations than explicitly expressed in the cost function is taken into account. For example, our cost function only models gradient index and selectivity indirectly, which means that we cannot guarantee Pareto optimality with respect to these metrics. On the other hand, our results show that manual forward plans are, in general, not Pareto optimal either and that the proposed inverse planner can often find plans that dominate them.

For convex multi-criteria optimization problems, such as ours, every choice of weights corresponds to a Pareto optimal solution. By specifying the weights before optimization, we have treated this problem as a single-criteria optimization problem~\citep{Ehrgott05multicriteriaoptimization}. However, every Pareto optimal point corresponds to the solution of the optimization problem for a set weights on the unit simplex, i.e. such that $w\geq 0$ and  $\sum_{i=1}^n w_i=1$. In other words, the formulation as a weighted sum does not restrict the set of optimal plans that we can obtain. In practice, our normalization is not ideal since the number of active points in each structure depends on a number of factors, including the volume and the sampling strategy. Consequently the exact weight settings are unlikely to generalize from one case to another even though they may be quite similar. We thus expect that the planner will have to adjust the weights a few times before arriving at a satisfactory solution.

Although not explored here, we anticipate that further computational gains are attainable by cleverly manipulating the dose rate matrix $\Phi(r)$ and adapting the matrix algebra accordingly\cite{Cho2001,Thieke2002,Zakarian2004}. We also expect that tuning or replacing the solver, or both, could shorten the optimization times further\cite{Olafsson2006}.

\section{Conclusions}

We present the first sector-duration optimization that uses linear programming. It uses a beam-on time (BOT) penalization tailored to the Gamma Knife, which reduces the BOT by a factor of 2--3 compared to the na\"{i}ve alternative. In addition to using linear programming, we describe two techniques that reduce the problem size and thus further reduce the solution time: dualization and representative subsampling. Dualization leads to an equivalent problem that can be solved 5--20 times faster than the primal one. With representative subsampling we refer to a stochastic sampling of positions, both in the interior and on the surface of relevant structures, to use in the optimization. We show that using 10\% of the original number of voxels is enough to generate plans for which the statistical fluctuations in coverage and selectivity are below 1\% while resulting in time-savings comparable to dualization.

The importance of different objectives, such as coverage, selectivity or beam-on time (BOT), are controlled by adjusting the weights of the corresponding terms in the cost function. In a comparison with 75 clinical plans we show that in 44 of these we can find plans that simultaneously have better selectivity, BOT and gradient index (with coverage close to 100\% in all cases) than the forward plans.

Treatment planning for Gamma Knife has always been highly interactive. Thanks to the combination of techniques to reduce the computational cost that we have presented, it becomes possible to fit sector-duration optimization---with all its benefits---into the clinical workflow. This is our main contribution.

\section*{Disclosure of Conflicts of Interest}
All authors are employed at Elekta Instrument AB, which holds several patents related to this area. The work presented may become part of a future commercial product.

\begin{acknowledgments}
The authors would like to thank Björn Somell and the anonymous reviewers for helpful comments and suggestions in the preparation of the manuscript. The Sharknado team---nuff said. The research was supported by the VINNOVA/ITEA3 project BENEFIT (grant 2014-00593).
\end{acknowledgments}

\appendix
\section{Explicit formulation of the primal problem}\label{appendix:explicit_primal}
Any linear programming problem can be written in the form
\begin{equation}
\begin{aligned}
& \underset{\vecx}{\text{minimize}}
& & \vecw^t\vecx \\
& \text{subject to}& & A\vecx = b\\
&&& \vecx\geq 0,\label{eq:standardLP}
\end{aligned}
\end{equation}
where $x\in\mathbb{R}^n$ are the optimization variables and $w$, $b$ and $A\in\mathbb{R}^{p \times n}$ define the objective and constraint functions of the problem.

We introduce auxiliary variables to rewrite the hinge and iBOT functions using linear programming:
if $y^+, y^-\geq 0$ and $D - \hat{D} = (y^+ - y^-)$, then $y^+\geq (D-\hat{D})_+$ and $y^-\geq (\hat{D}-D)_+$; if $\tau_i\geq \sum_{c=1}^3 t_{isc} $ for $s=1,\ldots,8$, then $\tau_i\geq \underset{s}{\max}\sum_{c=1}^3 t_{isc}$. We may then express the primal optimization problem \eqref{eq:costfcn} in standard form by making the following identifications:
\begin{equation}
\begin{aligned}
A &= \begin{pmatrix}
 \Phi_{\rm T} &-I&I&0&0&0&0&0&0&0\\
 \Phi_{\rm S} &0&0&-I&I&0&0&0&0&0\\
 \Phi_{\rm G} &0&0&0&0&-I&I&0&0&0\\
\Phi_{\rm O} &0&0&0&0&0&0&I&0&0\\
C &0&0&0&0&0&0&0&I&-T\\
  \end{pmatrix}, \\
x^t &=\left( \tOpt, y_\text{T}^+,y_\text{T}^-,y_\text{S}^+,y_\text{S}^-,y_\text{G}^+,y_\text{G}^-,p,q,\tau\right),\\
\vecw^t &=\left(0, 0, \frac{w_{\rm T}}{N_{\rm T} D_{\rm T}}, \frac{w_{\rm S}}{N_{\rm S} D_{\rm S}}, 0, \frac{w_{\rm G}}{N_{\rm G} D_{\rm G}}, 0, 0,0, \frac{w_{\rm BOT}}{D_\textrm{T}/\varphi_\text{cal}}\right),\\
b^t &= \left(D_{\rm T},D_{\rm S},D_{\rm G},D_{\rm O},0\right),
\end{aligned}
\label{eq:explicitPrimal}
\end{equation}
where

\begin{align}
C&=\begin{pmatrix}
1&1&1&0&0&0&\ldots&0&0&0\\
0&0&0&1&1&1&\ldots&0&0&0\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\
0&0&0&0&0&0&\ldots&1&1&1
\end{pmatrix}
= I_{N_\text{iso}}\otimes I_8\otimes (1, 1, 1)\in\mathbb{R}^{8N_\text{iso}\times 24N_\text{iso}},\\
T &=I_{N_\text{iso}}\otimes 1_{8\times 1}\in \mathbb{R}^{8N_\text{iso}\times N_\text{iso}},
\end{align}
and $\otimes$ denotes the Kronecker product.

\section{Explicit formulation of the dual problem}\label{appendix:explicit_dual}
Most modern linear programming solvers begin with a presolve step that intends to reduce the problem size \cite{Andersen1995}. However, for completeness and because we haven't encountered a solver that automatically detects the benefit of dualizing our problem, we will here carry out the dualization explicitly.

The dual problem corresponding to a linear programming problem on standard form, equation \eqref{eq:standardLP}, is\cite{Bertsimas1997,Boyd2004,Nocedal2006}
\begin{equation}
\begin{aligned}
& \underset{\nu}{\text{minimize}}
& & b^t\nu \\
& \text{subject to}& & A^t\nu+\vecw \geq 0,\label{eq:dualstandardLP}\\
\end{aligned}
\end{equation}

where the dual variable $\nu$ is the Lagrange multiplier for the linear constraints of the primal problem. By eliminating redundant variables and grouping constraints, our problem can be stated as
\begin{equation}
\begin{aligned}
& \underset{\tilde{\nu}}{\text{minimize}}
& & \rho^t\tilde{\nu} \\
& \text{subject to}& & B^t\tilde{\nu}\leq \sigma,\\
&& & \ell \leq \tilde{\nu} \leq \mu \,,\label{eq:ourDual}
\end{aligned}
\end{equation}
where we have introduced the following rescaled entities
\begin{equation}
\begin{aligned}
\tilde{\nu} &= (\frac{\nu_{\rm T}}{D_{\rm T}}, \frac{\nu_{\rm S}}{D_{\rm S}}, \frac{\nu_{\rm G}}{D_{\rm G}}, \frac{\nu_{\rm O}}{D_{\rm O}}, \frac{\nu_\text{iso}}{D_\textrm{T}/\varphi_\text{cal}})\,,\\
\varphi_{\rm T}&= \frac{\phi_{\rm T}}{D_{\rm T}},\,\varphi_{\rm S}= \frac{\phi_{\rm S}}{D_{\rm S}},\,\varphi_{\rm G}= \frac{\phi_{\rm G}}{D_{\rm G}},\,
\varphi_{\rm O}= \frac{\phi_{\rm O}}{D_{\rm O}},\, C_s = \frac{C}{D_\textrm{T}/\varphi_\text{cal}}.\\
\end{aligned}
\end{equation}
We may then express the dual optimization problem corresponding to equation \eqref{eq:explicitPrimal} by making the following identifications
\begin{equation}
\begin{aligned}
\rho^t &= \left(-1, 1, 1, 1, 0\right)\,,\\
B^t &= \begin{pmatrix}
    \varphi^t_{\rm T} & -\varphi^t_{\rm S} & -\varphi^t_{\rm G} & -\varphi^t_{\rm O} & -C_s^t  \\
    0 & 0 & 0 & 0 & T^t
  \end{pmatrix}\,,\\
\sigma^t &= (0,w_{\rm BOT})\,, \\
\ell^{\phantom t} &=  \left(0, 0, 0, 0, 0\right)\,,\\
\mu^{\phantom t} &= \left(\frac{w_{\rm T}}{N_{\rm T}},\frac{w_{\rm S}}{N_{\rm S}},\frac{w_{\rm G}}{N_{\rm G}},\infty,\infty \right)\,.
\end{aligned}
\end{equation}
The number of non-zero elements in $B$ is $\left(24(N_\text{T}+N_\text{S}+N_\text{G}+N_\text{O}) + 32\right)N_\text{iso}$.


TITLE: Is there a polynomial $g(x)$ such that $p \circ f=g \circ p$, where $f(x)=9x+30x^3+27x^5+9x^7+{{x}^{9}}$?
QUESTION [1 upvotes]: Let us consider a polynomial $f(x)=9x+30x^3+27x^5+9x^7+{{x}^{9}}$ over a field of characteristic $0$.
Set $p(x)=x^nh(x), n \geq 1$, where $h(x)$ can be any rational function.
I am trying to find out another polynomials $g(x)$ such that $p \circ f=g \circ p$. One more information, for any zero $\alpha$ of $f(x)$, $~~p(\alpha)$ is a root of $g(x)$.
If $h(x)=1$, then we need find a $g(x)$ such that $(f(x))^n=g(x^n)$.
I exclude the case $f = g, n=1$.
I have found the roots of $f(x)$ here in decimal form.
I thing this gives a hints about $g(x)$ because $f(\alpha)=0 \Rightarrow f(p(\alpha))=0$.
Edit: sorry for multiple edits but I hope this is the final one

REPLY [3 votes]: $$
\begin{array}{rcl}
p(x)&=&9x,\\
g(x)&=&\displaystyle \frac{x^9}{43046721}+\frac{x^7}{59049}+\frac{x^5}{243}+\frac{10 x^3}{27}+9 x.
\end{array}
$$
Upd. For the new version
$$
\begin{array}{rcl}
p(x)&=&x^2,\\
g(x)&=&x (x+3)^2 \left(x (x+3)^2+3\right)^2;
\end{array}
$$
$$
\begin{array}{rcl}
p(x)&=&x^4+4 x^2,\\
g(x)&=&x (x+3)^2 \left(x (x+3)^2+3\right)^2.
\end{array}
$$
$$
\begin{array}{rcl}
p(x)&=&x^2 \left(x^2+3\right)^2,\\
g(x)&=&x (x+3)^2 \left(x (x+3)^2+3\right)^2.
\end{array}
$$


TITLE: Notation involving the symmetric group $S_n$
QUESTION [2 upvotes]: I'm reading this paper by Hinich and he uses two notations involving the symmetric group $S_n$ that he doesn't clarify, so I assume that they are standard, but I don't know what they mean.
The first of them appears at the end of page 3:

Let $\mathscr{O}$ be an operad in a tensor category $\mathscr{A}$. Let $V$ be an $\mathbb{S}$-object in $\mathscr{A}$. The free $\mathscr{O}$-algebra generated by $V$ is defined to be
  $$
\Bbb{F}_{\mathscr{O}}(V)=\bigoplus_{n\geq 0}\mathscr{O}(n)\otimes_{S_n}V^{\otimes n}
$$
  with a canonical $\mathscr{O}$-algebra structure.

I think it doesn't make sense to take tensor product with respect to $S_n$ (as in $R$-modules with respect to $R$) so I guess it has to do with the $S_n$-equivariance of the action of the operad $\mathscr{O}$ on the algebra, but I'm not sure.
Then there is a dual definition

If $V$ is an $\mathbb{S}$-object in $\mathscr{A}$, the cofree coalgebra cogenerated by $V$ is defined to be
  $$
\Bbb{F}^*_{\mathscr{C}}(V)=\bigoplus_{n\geq 0}\left(\mathscr{C}(n)\otimes V^{\otimes n}\right)^{S_n}
$$

In this case I have no clue what it means. It cannot be the maps to $S_n$ since for $n=1$ it is used a few pages later (after equation (9)) that it is just $V$. Maybe it is just a direct sum of $n!$ copies, but that's just a guess.
I hope someone recognizes these notations. Thank you.

REPLY [0 votes]: I looked it up in Algebraic Operads by Loday and Vallette (chapter 5).
The first one is the tensor product taken under the action of $S_n$. On $\mathscr{O}(n)$ it is the action that has by definition of $\Bbb{S}$-object and in $V^{\otimes n}$ is a rearrangement of the coordinates.
The second one is the space of invariant elements under this action.


TITLE: Exterior tensor product of structure sheaves
QUESTION [7 upvotes]: I am reading the book "Fourier-Mukai transforms in algebraic geometry" by Daniel Huybrechts and to solve one of its questions, I came up to show that
$$\mathcal{O}_{X_1}\boxtimes\mathcal{O}_{X_2}=\mathcal{O}_{X_1\times X_2}$$
i.e.
$${\pi_{X_1}}^*\mathcal{O}_{X_1}\otimes{\pi_{X_2}}^*\mathcal{O}_{X_2}=\mathcal{O}_{X_1\times X_2}$$ where here $\pi_{X_1}$ (resp. $\pi_{X_2}$) is the projection from $X_1\times X_2\to X_1$ (resp. $X_1\times X_2\to X_2$) and I am looking at $\mathcal{O}_{X_1}$,  $\mathcal{O}_{X_2}$ and $\mathcal{O}_{X_1\times X_2}$ as objects in $D^b(X_1)$, $D^b(X_2)$ and $D^b(X_1\times X_2)$, respectively. My question is that is this equality true and if yes, why?

REPLY [9 votes]: Yes. The way you compute a derived pullback is to take a flat resolution and then pull that back, but the structure sheaf is trivially flat and pulls back to the structure sheaf.


TITLE: Using summation notation to solve Riemann Sum problems
QUESTION [0 upvotes]: I am learning how to write summation notation to solve Riemann Sum problems. I have a function
$f(x)=2x^3 -5x^2 +9x -5$ and I need to find the area under the curve with 8 intervals for $-4 \leq x \leq 12$ using midpoint Riemann Sum
I find the $\Delta x = 2$.
Then I use the midpoint formula $f(x_k) = a + \frac{1}{2}\Delta x + k \Delta x$ to find that $f(x_k) = -3 +2k$.
So I create the summation notation and plug the $f(x_k)$ value in place of $x$
$\sum_{k=0}^{n=7} (2(-3+2k)^3-5(-3+2k)^2+9(-3+2k)-5))2$
When I plug this into https://www.mathsisfun.com/numbers/sigma-calculator.html I get the answer 8,880 instead of the correct answer 7,648. Could some tell me where I am going wrong?

REPLY [1 votes]: What you should have added is$$\sum_{k=0}^72\times f(-3+2k).$$Try it. You'll get $7\,648$.


TITLE: Find all subrings of a ring
QUESTION [3 upvotes]: Given a finite ring, e.g $\mathbb{Z}{_{24}}$, how can I find all of its subrings?
I have tried to think about it couldn't reach any idea.
Thanks.

REPLY [3 votes]: Hint: Start by finding all additive subgroups. Which of those are closed under multiplication?


TITLE: Finding the boundary of a subset of $\mathbb{R}$
QUESTION [2 upvotes]: Let $\mathbb{Q}=\{q_1, q_2,...\}$ and $S=\bigcup_{k\in\mathbb{N}}{N_{1/2^k}}{(q_k)}$ (where $N_{1/2^k}(q_k)$ is the open neighborhood around $q_k$)  be a subset of $\mathbb{R}$.
The questions asks to find boundary of $S$.
I know that the boundary of a set is the intersection of its closure and the closure of its compliment. Since we are examining shrinking neighborhoods around the elements in $\mathbb{Q}$, and since $\mathbb{Q}$ is dense in $\mathbb{R}$, is the boundary of $S$ just $\mathbb{R}$ itself? If this isn't correct, could someone point me in the right direction? Many thanks.

REPLY [1 votes]: Answer. $\,\partial S=\mathbb R\smallsetminus S$.
By definition, $\partial S=\overline{S}\cap\overline{\mathbb R\smallsetminus S}$,
and $\mathbb R\smallsetminus S$ is closen, then  $\partial S=\overline{S}\cap{\mathbb R\smallsetminus S}$. It suffices to show that
$$
x\in\mathbb R\smallsetminus S\quad\Longrightarrow\quad x\in\overline{S}.
$$
But is $x\not\in S$, then $x$ is irrational, and it can be approximated by rationals, and hence $x$ is a limit point of $S$, i.e., $x\in\overline{S}$.


TITLE: If $\lim{a_n}=0$ then there exists $a_{k_n}$ such that $\lim n a_{k_n}=0$ in $\mathbb R$
QUESTION [4 upvotes]: If $\lim{a_n}=0$    then there exists $a_{k_n}$ such that $\lim n a_{k_n}=0$  in $\mathbb R$
Attempt.
First I wrongly thought and divided given $\epsilon$ into $n$ parts and then sum. But it is wrong because $ n $ is not constant.
Second I tried to play with partial sums and their difference for example $2n a_n<\sum S_{2n}-S_n....$ but I realized I didnot have if it is convergence  or divergence series with $a_n$
Third I consider some kind of subsequences that analog to for example Cauchy s $2^n$ series... or consider the other subsequence functions but I failed..
How can I start? Can you give me a hint please?

REPLY [6 votes]: The other answers have answered your question (which amounts to asking: why, if $a_n\to0$, must there be a subsequence that vanishes faster than $n^{-1}$?).
But there is an obvious generalization, which shows there is nothing special about the factor $n$ in your problem. Informally, the point is that if $a_n$ tends to zero, we can always select a subsequence that vanishes "as fast as we want" -- faster than the reciprocal of any sequence that diverges to infinity. The proof idea is exactly the same.

Claim. Suppose $a_n\to 0$ and $b_n\to\infty$ with $b_n\neq 0$ for all $n$. Then there exists a subsequence $a_{k_n}$ such that $b_na_{k_n}\to0$.

Proof. Define
$$k_n=\min\left\{i\,|\,i\in\mathbb{N}, |a_i|<b_n^{-2}\right\}$$
This definition is well-defined because $a_n\to0$ (and every set of natural numbers has a least element).
Then
$$0\leq|b_na_{k_n}|<|b_n|b_n^{-2}=|b_n^{-1}|$$
The result follows by the squeeze theorem.
Note that squaring is not essential. We could use $b_n^{-\alpha}$ for $\alpha>1$.


TITLE: Showing that $H_A\simeq H_{A'}$ implies $A\simeq A'$
QUESTION [0 upvotes]: I'm trying to solve this exercise:

So we are given that for all objects $B$, there is a natural isomorphism $$H_A(B)\simeq H_{A'}(B)$$
Let's write this isomorphism as $f\mapsto \bar f$. The commutativity of the square tells us $$\overline{f\circ\phi}=\bar f\circ\phi$$ where $\phi:B'\to B$ is an arrow in $\mathscr A$. I don't see how to use that. I don't even see how to show that there exists an arrow $A\to A'$ (or $A'\to A$).

REPLY [1 votes]: You do need the definition that $H_A(B) = \operatorname{Hom}(B, A)$. In particular, this means that $Id_A \in H_A(A)$, so this corresponds to some $f: A \to A'$ under the isomorphism $H_A(A) \cong H_{A'}(A)$. Similarly, $Id_{A'} \in H_{A'}(A')$ corresponds to some $g: A' \to A$ in $H_A(A')$. The claim is now that $f$ is an isomorphism, with inverse $g$.
To prove this, the following diagram may be helpful:

The details of this can be found in the spoiler block below, but it is a good exercise to write these out yourself.

 It just comes down to chasing through the diagram. If we start in the top left with $Id_A \in H_A(A)$, then by definition this corresponds to $f \in H_{A'}(A)$, which gets sent to $fg = H_{A'}(g)(f) \in H_{A'}(A')$. Going the other way, $Id_A \in H_A(A)$ is first sent to $g = H_A(g)(Id_A) \in H_A(A')$. and by definition $g$ corresponds to $Id_{A'} \in H_{A'}(A')$ under the isomorphism. By naturality of the isomorphism, these two should be the same, so $fg = Id_{A'}$. A similar reasoning starting at $Id_{A'} \in H_{A'}(A')$ and using naturality with respect to $f$ instead, we also conclude $gf = Id_A$. So indeed $f$ is an isomorphism with inverse $g$.


TITLE: Integrate $\int_c \frac{dz}{z^2 + 4}$ and $\int_c \sqrt z dz $
QUESTION [1 upvotes]: (a) Integrate $\displaystyle \int_c \frac{dz}{z^2 + 4}$ along the line $x + y = 1$ in the direction of increasing $x$.
(b) Evaluate $\displaystyle \int_{-2 - 2 \sqrt 3 i}^{-2 + 2 \sqrt 3i}  \sqrt z dz $ along the straight line path if we choose that branch of $z^{1/2}$ such that $z^{1/2} = 1$ for $z = 1$

The answers are given as $\pi/2$ and $32/3$. For first I tried to substitue $x = t, y = 1 - t$ and integrated from $-\infty \to \infty $ but I can't get the answer.
For second I tried to integrate it directly, as well as putting $z = 4 e^{i (\theta + 2 \pi)}$ but can't seem to get the answer.
Help!! Thanks!!

REPLY [2 votes]: The second problem is difficult because if you blindly evaluate the integral without being sensitive to the branch cut across the negative real axis (necessary because of the demand that $\sqrt{1} = 1$), you get
$$\int_{4 e^{-i 2 \pi/3}}^{4 e^{i 2 \pi/3}} dz \sqrt{z} = \frac{2}{3} 4^{3/2} \left (e^{i \pi} - e^{-i \pi} \right ) = 0$$
To do this integral correctly, you need to add $2 \pi$ to the argument of the upper integration limit because the transition across the branch cut introduces that discontinuity.  Therefore we have
$$\int_{4 e^{-i 2 \pi/3}}^{e^{i (2 \pi/3 + 2 \pi)}} dz \sqrt{z} =  \frac{2}{3} 4^{3/2} \left (e^{i( \pi + 3 \pi)} - e^{-i \pi} \right ) = 2 \frac{2}{3} 2^3 = \frac{32}{3}$$


TITLE: checking slope = $0$ at a point for a function using $\epsilon $, $\delta $ definition
QUESTION [4 upvotes]: From the continuity definition, a function is continuous at a point $a$ if  : $$\forall \epsilon \gt 0 \exists \delta \gt 0  : |x-a| \lt \delta \implies |f(x)-f(a)|\lt \epsilon$$
If I change the order of quantifiers like below do I get a definition for checking slope = $0$ at $x = a$ ?
$$\exists \delta \gt 0   \forall \epsilon \gt 0 : |x-a| \lt \delta \implies |f(x)-f(a)|\lt \epsilon$$

If my interpretation of second quantifier order is flawed, could you please tell me how to interpret it correctly in this context ? I realized that order matters by going through other mse posts already.

REPLY [2 votes]: The order of the quantifies matters a lot. The statement
$$\exists \delta \gt 0, \forall\epsilon \gt 0 : |x-a| \lt \delta \implies |f(x)-f(a)|\lt \epsilon$$
implies that there exist an interval around $a$ where $f(x)$ is constant. If we change the order of the quantifiers to
$$\forall \delta \gt 0, \exists\epsilon \gt 0 : |x-a| \lt \delta \implies |f(x)-f(a)|\lt \epsilon$$
then for example any function which is bounded will satisfy it since if $|f(x)| < M$ then take $\epsilon = 2M$ to get $|f(x)-f(a)| < \epsilon$ for all $\delta$.