diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx
index 98272b9..663d9ed 100644
--- a/lepaper/arrayscat.lyx
+++ b/lepaper/arrayscat.lyx
@@ -260,12 +260,12 @@ status open
 
 
 \begin_inset FormulaMacro
-\newcommand{\particle}{\mathrm{\Theta}}
+\newcommand{\medium}{\Theta}
 \end_inset
 
 
 \begin_inset FormulaMacro
-\newcommand{\medium}{\thespace\backslash\particle}
+\newcommand{\mezikuli}[3]{\Theta_{#1,#2}\left(#3\right)}
 \end_inset
 
 
@@ -355,7 +355,7 @@ status open
 
 
 \begin_inset FormulaMacro
-\newcommand{\closedball}[2]{B_{#1}#2}
+\newcommand{\closedball}[2]{\overline{B_{#1}\left(#2\right)}}
 \end_inset
 
 
@@ -566,7 +566,17 @@ The (somewhat underrated) T-matrix multiple scattering method (TMMSM) can
 Here we extend the method to infinite periodic structures using Ewald-type
  lattice summation, and we exploit the possible symmetries of the structure
  to further improve its efficiency.
- (SHOULD I MENTION ALSO THE CROSS SECTION FORMULAS IN THE ABSTRACT?)
+ 
+\begin_inset Marginal
+status open
+
+\begin_layout Plain Layout
+Should I mention also the cross sections formulae in abstract / intro?
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Abstract
@@ -736,11 +746,6 @@ Maybe put the numerical results separately in the end.
 TODOs
 \end_layout
 
-\begin_layout Itemize
-Consistent notation of balls.
- How is the difference between two cocentric balls called?
-\end_layout
-
 \begin_layout Itemize
 It could be nice to include some illustration (example array) to the introductio
 n.
@@ -752,26 +757,12 @@ Maybe mention that in infinite systems, it can be also much faster than
  other methods.
 \end_layout
 
-\begin_layout Itemize
-Translation operators: rewrite in sph.
- harm.
- convention independent form.
-\end_layout
-
 \begin_layout Itemize
 Truncation notation.
 \end_layout
 
 \begin_layout Itemize
-Example results!
-\end_layout
-
-\begin_layout Itemize
-Figures.
-\end_layout
-
-\begin_layout Itemize
-Concrete comparison with other methods.
+Example results and benchmarks with BEM; figures!
 \end_layout
 
 \begin_layout Itemize
diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx
index 6d6e187..5e83f82 100644
--- a/lepaper/finite.lyx
+++ b/lepaper/finite.lyx
@@ -138,16 +138,40 @@ Single-particle scattering
 \begin_layout Standard
 In order to define the basic concepts, let us first consider the case of
  EM radiation scattered by a single particle.
- We assume that the scatterer lies inside a closed sphere 
-\begin_inset Formula $\particle$
+ We assume that the scatterer lies inside a closed ball 
+\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
 \end_inset
 
-, the space outside this volume 
-\begin_inset Formula $\medium$
+ of radius 
+\begin_inset Formula $R^{<}$
 \end_inset
 
- is filled with an homogeneous isotropic medium with relative electric permittiv
-ity 
+ and center in the origin of the coordinate system (which can be chosen
+ that way; the natural choice of 
+\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
+\end_inset
+
+ is the circumscribed ball of the scatterer) and that there exists a larger
+ open cocentric ball 
+\begin_inset Formula $\openball{R^{>}}{\vect 0}$
+\end_inset
+
+, such that 
+\begin_inset Marginal
+status open
+
+\begin_layout Plain Layout
+Is there a word for this?
+\end_layout
+
+\end_inset
+
+ the (non-empty) volume 
+\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}=\openball{R^{>}}{\vect 0}\setminus\closedball{R^{<}}{\vect 0}$
+\end_inset
+
+ is filled with a homogeneous isotropic medium with relative electric permittivi
+ty 
 \begin_inset Formula $\epsilon(\vect r,\omega)=\epsbg(\omega)$
 \end_inset
 
@@ -285,7 +309,16 @@ where
 \end_inset
 
  are the regular spherical Bessel function and spherical Hankel function
- of the first kind, respectively, as in [DLMF §10.47], and 
+ of the first kind, respectively, as in 
+\begin_inset CommandInset citation
+LatexCommand cite
+after "§10.47"
+key "NIST:DLMF"
+literal "false"
+
+\end_inset
+
+, and 
 \begin_inset Formula $\vsh{\tau}lm$
 \end_inset
 
@@ -307,7 +340,16 @@ In our convention, the (scalar) spherical harmonics
 \begin_inset Formula $\ush lm$
 \end_inset
 
- are identical to those in [DLMF 14.30.1], i.e.
+ are identical to those in 
+\begin_inset CommandInset citation
+LatexCommand cite
+after "14.30.1"
+key "NIST:DLMF"
+literal "false"
+
+\end_inset
+
+, i.e.
 \begin_inset Formula 
 \[
 \ush lm=\left(\frac{\left(l-m\right)!\left(2l+1\right)}{4\pi\left(l+m\right)!}\right)^{\frac{1}{2}}e^{im\phi}\dlmfFer lm\left(\cos\theta\right)
@@ -319,8 +361,16 @@ where importantly, the Ferrers functions
 \begin_inset Formula $\dlmfFer lm$
 \end_inset
 
- defined as in [DLMF §14.3(i)] do already contain the Condon-Shortley phase
- 
+ defined as in 
+\begin_inset CommandInset citation
+LatexCommand cite
+after "§14.3(i)"
+key "NIST:DLMF"
+literal "false"
+
+\end_inset
+
+ do already contain the Condon-Shortley phase 
 \begin_inset Formula $\left(-1\right)^{m}$
 \end_inset
 
@@ -418,7 +468,7 @@ The regular VSWFs
 \begin_inset Formula $\vswfrtlm{\tau}lm\left(k\vect r\right)$
 \end_inset
 
- constitute a basis for solutions of the Helmholtz equation 
+ would constitute a basis for solutions of the Helmholtz equation 
 \begin_inset CommandInset ref
 LatexCommand eqref
 reference "eq:Helmholtz eq"
@@ -429,16 +479,17 @@ noprefix "false"
 \end_inset
 
  inside a ball 
-\begin_inset Formula $\openball 0{R^{>}}$
+\begin_inset Formula $\openball{R^{>}}{\vect 0}$
 \end_inset
 
  with radius 
 \begin_inset Formula $R^{>}$
 \end_inset
 
- and center in the origin; however, if the equation is not guaranteed to
- hold inside a smaller ball 
-\begin_inset Formula $B_{0}\left(R\right)$
+ and center in the origin, were it filled with homogeneous isotropic medium;
+ however, if the equation is not guaranteed to hold inside a smaller ball
+ 
+\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
 \end_inset
 
  around the origin (typically due to presence of a scatterer), one has to
@@ -447,7 +498,7 @@ noprefix "false"
 \end_inset
 
  to have a complete basis of the solutions in the volume 
-\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
+\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}=\openball{R^{>}}{\vect 0}\setminus\closedball{R^{<}}{\vect 0}$
 \end_inset
 
 .
@@ -469,11 +520,11 @@ The single-particle scattering problem at frequency
 \end_inset
 
  can be posed as follows: Let a scatterer be enclosed inside the ball 
-\begin_inset Formula $B_{0}\left(R\right)$
+\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
 \end_inset
 
  and let the whole volume 
-\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
+\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
 \end_inset
 
  be filled with a homogeneous isotropic medium with wave number 
@@ -482,7 +533,7 @@ The single-particle scattering problem at frequency
 
 .
  Inside 
-\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
+\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
 \end_inset
 
 , the electric field can be expanded as
@@ -504,7 +555,7 @@ doplnit frekvence a polohy
 \end_inset
 
 If there was no scatterer and 
-\begin_inset Formula $B_{0}\left(R_{<}\right)$
+\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
 \end_inset
 
  was filled with the same homogeneous medium, the part with the outgoing
@@ -513,7 +564,7 @@ If there was no scatterer and
 \end_inset
 
  due to sources outside 
-\begin_inset Formula $\openball 0R$
+\begin_inset Formula $\openball{R^{>}}{\vect 0}$
 \end_inset
 
  would remain.
@@ -670,7 +721,17 @@ literal "false"
 \end_inset
 
 -matrix results wrong; we found and fixed the bug and from upstream version
- xxx onwards, it should behave correctly.
+ xxx
+\begin_inset Marginal
+status open
+
+\begin_layout Plain Layout
+Not yet merged to upstream.
+\end_layout
+
+\end_inset
+
+ onwards, it should behave correctly.
  
 \end_layout
 
@@ -689,8 +750,25 @@ The magnitude of the
 \begin_inset Formula $T$
 \end_inset
 
--matrix of a bounded scatterer is a compact operator [REF???], so from certain
- multipole degree onwards, 
+-matrix of a bounded scatterer is a compact operator 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "ganesh_convergence_2012"
+literal "false"
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO
+\end_layout
+
+\end_inset
+
+, so from certain multipole degree onwards, 
 \begin_inset Formula $l,l'>L$
 \end_inset
 
@@ -725,16 +803,6 @@ The magnitude of the
 \end_inset
 
  will also be negligible.
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-TODO when it will not be negligible
-\end_layout
-
-\end_inset
-
-
 \end_layout
 
 \begin_layout Standard
@@ -761,7 +829,7 @@ literal "false"
 \end_inset
 
  by requiring that 
-\begin_inset Formula $\delta\gtrsim\left(nR\right)^{L}/\left(2L+1\right)!!$
+\begin_inset Formula $\delta\gg\left(nR\right)^{L}/\left(2L+1\right)!!$
 \end_inset
 
 , where 
@@ -886,7 +954,7 @@ literal "true"
 
 .
  Let the field in 
-\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
+\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
 \end_inset
 
  have expansion as in 
@@ -901,11 +969,11 @@ noprefix "false"
 
 .
  Then the net power transported from 
-\begin_inset Formula $B_{0}\left(R\right)$
+\begin_inset Formula $\openball{R^{<}}{\vect 0}$
 \end_inset
 
  to 
-\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
+\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
 \end_inset
 
  via by electromagnetic radiation is
@@ -917,7 +985,7 @@ P=\frac{1}{2k^{2}\eta_{0}\eta}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\ri
 \end_inset
 
 In realistic scattering setups, power is transferred by radiation into 
-\begin_inset Formula $B_{0}\left(R\right)$
+\begin_inset Formula $\openball{R^{<}}{\vect 0}$
 \end_inset
 
  and absorbed by the enclosed scatterer, so 
@@ -1087,25 +1155,15 @@ If the system consists of multiple scatterers, the EM fields around each
 \end_inset
 
  be an index set labeling the scatterers.
- We enclose each scatterer in a ball 
-\begin_inset Formula $B_{\vect r_{p}}\left(R_{p}\right)$
+ We enclose each scatterer in a closed ball 
+\begin_inset Formula $\closedball{R_{p}}{\vect r_{p}}$
 \end_inset
 
  such that the balls do not touch, 
-\begin_inset Formula $B_{\vect r_{p}}\left(R_{p}\right)\cap B_{\vect r_{q}}\left(R_{q}\right)=\emptyset;p,q\in\mathcal{P}$
+\begin_inset Formula $\closedball{R_{p}}{\vect r_{p}}\cap\closedball{R_{q}}{\vect r_{q}}=\emptyset;p,q\in\mathcal{P}$
 \end_inset
 
-, 
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-TODO bacha, musejí být uzavřené!
-\end_layout
-
-\end_inset
-
-so there is a non-empty volume
+, so there is a non-empty volume
 \begin_inset Note Note
 status open
 
@@ -1116,12 +1174,20 @@ jaksetometuje?
 \end_inset
 
  
-\begin_inset Formula $\openball{\vect r_{p}}{R_{p}^{>}}\backslash B_{\vect r_{p}}\left(R_{p}\right)$
+\begin_inset Formula $\mezikuli{R_{p}}{R_{p}^{>}}{\vect r_{p}}$
 \end_inset
 
- around each one that contains only the background medium without any scatterers.
- Then the EM field inside each such volume can be expanded in a way similar
- to 
+ around each one that contains only the background medium without any scatterers
+ (we assume that all the volume outside 
+\begin_inset Formula $\bigcap_{p\in\mathcal{P}}\closedball{R_{p}}{\vect r_{p}}$
+\end_inset
+
+ is filled with the same background medium).
+ Then the EM field inside each 
+\begin_inset Formula $\mezikuli{R_{p}}{R_{p}^{>}}{\vect r_{p}}$
+\end_inset
+
+ can be expanded in a way similar to 
 \begin_inset CommandInset ref
 LatexCommand eqref
 reference "eq:E field expansion"
@@ -1135,7 +1201,7 @@ noprefix "false"
 \begin_inset Formula 
 \begin{align}
 \vect E\left(\omega,\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoeffptlm p{\tau}lm\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)+\outcoeffptlm p{\tau}lm\vswfouttlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)\right),\label{eq:E field expansion multiparticle}\\
- & \vect r\in\openball{\vect r_{p}}{R_{p}^{>}}\backslash B_{\vect r_{p}}\left(R_{p}\right).\nonumber 
+ & \vect r\in\mezikuli{R_{p}}{R_{p}^{>}}{\vect r_{p}}.\nonumber 
 \end{align}
 
 \end_inset
@@ -1447,10 +1513,10 @@ reference "eq:translation operator"
  below.
  For singular (outgoing) waves, the form of the expansion differs inside
  and outside the ball 
-\begin_inset Formula $\openball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}:$
+\begin_inset Formula $\closedball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}$
 \end_inset
 
- 
+:
 \begin_inset Formula 
 \begin{eqnarray}
 \vswfouttlm{\tau}lm\left(k\left(\vect r-\vect r_{1}\right)\right) & = & \begin{cases}
@@ -1669,8 +1735,17 @@ and analogously the elements of the singular operator
 
 \end_inset
 
-where the constant factors in our convention read (TODO CHECK ONCE AGAIN
- CAREFULLY FOR POSSIBLE PHASE FACTORS FACTORS)
+where the constant factors in our convention read 
+\begin_inset Marginal
+status open
+
+\begin_layout Plain Layout
+TODO check once again carefully for possible phase factors.
+\end_layout
+
+\end_inset
+
+
 \begin_inset Note Note
 status collapsed
 
@@ -1979,8 +2054,8 @@ literal "false"
  derives only the extinction cross section formula.
  Let us re-derive it together with the many-particle scattering and absorption
  cross sections.
- First, let us take a ball circumscribing all the scatterers at once, 
-\begin_inset Formula $\openball R{\vect r_{\square}}\supset\particle$
+ First, let us take a ball containing all the scatterers at once, 
+\begin_inset Formula $\openball R{\vect r_{\square}}\supset\bigcup_{p\in\mathcal{P}}\closedball{R_{p}}{\vect r_{p}}$
 \end_inset
 
 .
@@ -2010,8 +2085,7 @@ where
 \begin_inset Formula $\outcoeffp{\square}$
 \end_inset
 
- using the translation operators (REF!!!) and use the single-scatterer formulae
- 
+ using the translation operators and use the single-scatterer formulae 
 \begin_inset CommandInset ref
 LatexCommand eqref
 reference "eq:extincion CS single"
@@ -2113,8 +2187,18 @@ noprefix "false"
 
 where only the last expression is suitable for numerical evaluation with
  truncated matrices, because the previous ones contain a translation operator
- right next to an incident field coefficient vector (see Sec.
- TODO).
+ right next to an incident field coefficient vector
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+(see Sec.
+ TODO)
+\end_layout
+
+\end_inset
+
+.
  Similarly, 
 \begin_inset Formula 
 \begin{align}
diff --git a/lepaper/intro.lyx b/lepaper/intro.lyx
index 2ef2fa5..a55bf4c 100644
--- a/lepaper/intro.lyx
+++ b/lepaper/intro.lyx
@@ -175,10 +175,29 @@ superposition
 \end_inset
 
 -matrix method
+\begin_inset Marginal
+status open
+
+\begin_layout Plain Layout
+a.k.a.
+ something else?
+\end_layout
+
+\end_inset
+
+
 \emph default
- (TODO a.k.a something; refs??), and it has been implemented previously for
- a limited subset of problems (TODO refs and list the limitations of the
- available).
+, and it has been implemented previously for a limited subset of problems
+\begin_inset Marginal
+status open
+
+\begin_layout Plain Layout
+Refs; list the limitations of available codes?
+\end_layout
+
+\end_inset
+
+.
  
 \begin_inset Note Note
 status open
@@ -237,18 +256,46 @@ We hereby release our MSTMM implementation, the
 QPMS Photonic Multiple Scattering
 \emph default
  suite, as free software under the GNU General Public License version 3.
- (TODO refs to the code repositories.) QPMS allows for linear optics simulations
- of arbitrary sets of compact scatterers in isotropic media.
+ 
+\begin_inset Marginal
+status open
+
+\begin_layout Plain Layout
+TODO refs to the code repositories once it is published.
+\end_layout
+
+\end_inset
+
+ QPMS allows for linear optics simulations of arbitrary sets of compact
+ scatterers in isotropic media.
  The features include computations of electromagnetic response to external
  driving, the related cross sections, and finding resonances of finite structure
 s.
  Moreover, it includes the improvements covered in this paper, enabling
  to simulate even larger systems and also infinite structures with periodicity
  in one, two or three dimensions, which can be e.g.
- used for quickly evaluating dispersions of such structures, and also their
- topological invariants (TODO).
+ used for quickly evaluating dispersions of such structures
+\begin_inset Marginal
+status open
+
+\begin_layout Plain Layout
+And also their topological invariants (TODO)?
+\end_layout
+
+\end_inset
+
+.
  The QPMS suite contains a core C library, Python bindings and several utilities
  for routine computations.
+\begin_inset Marginal
+status open
+
+\begin_layout Plain Layout
+Such as?
+\end_layout
+
+\end_inset
+
  
 \begin_inset Note Note
 status open