Unify balls notations, some refs etc.
Former-commit-id: 69e5ae075b639a6aed4988b9e8801947017026a5
This commit is contained in:
Marek Nečada
parent
2b031d43da
commit
46b651d97f
|
|
@ -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
|
||||
|
|
|
|||
|
|
@ -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}
|
||||
|
|
|
|||
|
|
@ -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
|
||||
|
|
|
|||
Loading…
Reference in New Issue